First let me say that I'm not a newbie to this. I've been gardening for 15 years, so I've been around the block.
I have tried wood chip/bark mulch. Wood chips or bark mulch, about 4" to 6" worth. And then you renew it every few years by dumping a little more on top as it breaks down.
If not by the 2nd year it has broken down sufficiently to germinate weed seeds (usually wild lettuce) and the 2nd year it is no better than dirt. Even with landscape cloth underneath it, it would have to be removed and replaced every other year.
There is no cloth or plastic currently in existence on Planet Earth that is available to home consumers not affiliated with NASA or the MIT research labs that will not eventually break down under exposure to UV light, heat, and moisture.
And that won't eventually have holes popped in it as it breaks down and becomes brittle.
Gravel, pebbles, and rocks are stable.
You can put landscape rocks over weedstop fabric, and then the weedstop will break down after about 10 to 20 years and begin flapping fragments in the breeze, and you have to break it all down and do it over again.
The reason why people put mulches, either organic or inorganic, on top of plastic or cloth weedstop fabrics is precisely so they'll be protected from UV light and from the weather. Unprotected weedstop will last maybe a year. Protected, it's good for maybe 20 years, tops. But then sooner or later, it degrades and fragments.
This isn't how Nature, and Gardening, work.
Even concrete and asphalt get weeds growing up through cracks. You can put down gravel, pebbles, or rocks without anything under it and then periodically spot treat for weeds, either with Roundup or a propane weed torch.
Or you can put down bark or wood chips, and simply add more on top every few years.
If you omit the weedstop, you don't have to remove the mulch as it breaks down--you simply add more on top.
When you start seeing weeds germinating and growing, then you know it's not deep enough, so you add more. I keep about 3" to 4" of bark mulch on a formerly infested front flowerbed, and the only weeds I have to pull is the grass that sends runners over the double layer of buried brick. I add more mulch on top, an inch or two, about every 3 to 4 years.
I would scrape branches in areas that you think are dead to ensure they are not green underneath. Confirm the dead branches and cut them out so they can be filled in with new growth. Also, if they snap easily it's generally a safe bet it is dead.
Removing large dead branches isn't ideal because it can give the plant a funny look for some time, but it will fill in with fertilizer and water especially. In most cases, better off to just get it over with if it's truly dead.
One additional, but more drastic approach is if it was newly planted, you could dig it up and turn the new face of the plant where you want it, hiding the bare spots a bit. Keep in mind if you do this, watering needs increase.
Either way, turning some compost in the surround soil is always a good idea as well!
A French drain is one way to solve the problem, however I think you would lose that tree in the process of installing it. It's about as simple as you say but you should be back filling with river rock or larger gravel. The pipe needs holes. The trench should be at least a few feet deep and wide.
Working around those big roots could do a lot of damage, possibly killing the tree. Then you've got another problem on your hands. This is the most expensive and effective solution.
Another solution would be planting plants that like lots of water or possibly another tree. You would lose that area as functioning space but you'd also have some nice foliage to glance at. Kinda like a little wild garden.
The water has to have somewhere to go. Actually you should go about 8 inches deep, lay the trench with a landscaping fabric, lay the pipe on top of the fabric and cover with a clean washed gravel.
Then wrap the fabric over top of it. Then put dirt on top of that and replant.
Another way would be to bring in 5 yards of topsoil and raise the grade of the area to push that water else where. Unfortunately it looks like it would be sent directly into your neighbors back yard.
Not sure if they'd be too fond of that.
I've got one other way that I've seen work and that's simply laying sod directly on top of the existing ground. It raises the grade just a bit and soaks up a fair amount of water too.
We had a "bumpy lawn" and tried everything. You'll probably end up needing to 'top-dress' the areas affected by these 'bumps'. Top dressing is a term used to describe a mixture of washed sand (clean) and, perhaps, some compost.
You'll need to get a bunch of it, broadcast it over the affected area, rake it out, fill all the voids by any means necessary, broom the sand sitting on top into the void spaces between the grass, and then water it all in.
The grass will eventually come through the top dressing and you'll have a nice smooth lawn.
Another option is to water it really well, then pack down the affected area with a lawn roller. But this can backfire.
If your board or whoever will let you build a border, this would be your solution.
To add 4" of soil or mulch, you'll need to install a border of some kind around the entire hill, either of landscape timbers or of pavers, making it into a raised bed around the tree, to hold the soil or mulch in place against washing downhill every time it rains.
Same thing for gravel which needs something to hold it in place, otherwise time and traffic will slowly but surely move it all down the hill (and out into the street and parking lot), leaving the area under the tree bare again.
But if they won't let you build the border, then you can't cover up the roots, and then you can't make it into a place for people to congregate because they will trip over the exposed roots. You can't even put some benches directly on the ground or around the tree trunk, because people will trip over the roots.
You can't lay a path on top of the roots, so it would need to be quite far out from the tree, away from the roots, and thus would be quite a large path, whether it went anywhere or just went around in a circle.
If you're going to cover the roots with 4" of anything, no matter what the diameter of the space you're going to cover, whether a tiny border just around the tree trunk, or a huge border around the entire hill, it needs to have a paver or landscape timbers border constructed around the perimeter to hold it all in place. It will be a terrace, elevated above the surrounding parking lot by about 6".
People may trip over it, or drive over it, so your board needs to check their insurance.
You cannot restore the grass because, although the area is heavily shaded by the tree and you could plant a shade mixture there, the amount of traffic that has worn it away to mud as people take shortcuts across it would mitigate against any grass taking hold there. Anything you plant will simply get trampled on again, because people don't change their habits.
They're going to cut that corner no matter what's growing there.
The only way you'd be able to restore lawn there would be if you were to build a fence or hedge around the entire corner, or if you were able to build a border to hold the necessary soil in place, which border--theoretically at least--would also serve to warn drivers not to drive there.
Generally a fence or hedge needs to be at least 3 to 4 feet high to discourage people from simply stepping over it and taking their usual shortcut, and to make it obvious to drivers.
You could plant a ground cover there without adding any soil, but it would hide the tree roots, and people would trip over them as they cut across the ground cover. So, again, you'd need a pedestrian-proof fence. Or else a border, and some soil.
I don't understand where a path would go. Just around the tree in a big circle? Generally a path goes from Point A to Point B, not simply around in a circle.
You could potentially put a pathway around it, but due to the high level of pedestrian traffic and the potential for litigation, a flagstone pathway would probably not be a good idea, as even well-positioned flagstones can catch a heel and trip someone, so that leaves a poured concrete sidewalk, which may be a little beyond your skill level.
Quite frankly, I think you've been given very difficult task, and you are not wrong when you are feeling out of your depth.
This is a project that would be a challenge even for a semi-experienced DIY homeowner to implement on his own property.
The people who suggested this to you don't know what to do with it, either, and they're hoping you can come up with something. There would be absolutely no shame in a high school kid saying, "Uh, I think I'm in over my head here", and passing on the project.
Part of being smart is knowing when to back away.
If you just need a project, one thing you could do would be a much smaller version of the raised bed border, just around the tree trunk, about 3 feet wide out from the trunk.
This is standard around trees, it's easy to build with pavers or bricks, and you simply pour a couple inches of mulch into it and you're done. It doesn't address the entire corner with the hill and the roots, but it does make the tree itself look nice.
You could then place a bench or two on top of the soil or mulch which is being held in place by the landscape timbers or pavers.
Benches don't necessarily need a pathway unless someone's going to make an issue of wheelchair access.
I'm looking at a couple things I found when thinking over the tarp idea, mostly UV resistant commercial grade groundcovers.
One of them is inexpensive concrete stones are one of the better solutions to muddy areas. They're fairly indestructible and they stop most weeds.
In theory you could cut 7 mil tarp into 2' wide strips and just sort of staple / landscape stake it into the ground, then cover the entire thing in mulch, and weeds would have a difficult time growing into it, and it would survive being hit with a shovel. I think it would be both cheaper and more durable than landscape fabric.
I mean ideally you'd dig the whole path out and have it filled with gravel, but that's a lot more expensive and digging that long of a path would be an insane undertaking - you'd have to rent a digger.
Yet if you're putting mulch over a tarp the tarp isn't doing much. Most weeds won't grow through 3-4 inches of mulch, but rather grow from seeds on the surface.
I know that works great on a flowerbed or around trees or other large plants, and I've used it myself. The problem with using bark or wood chips on high traffic walk paths is that it breaks down faster than it does in a place where it rarely/never gets walked on.
Last spring I removed the last path that had bark/wood mulch on it (this was a mix of wood from the wood chipper, and stuff that was raked up from the area I split firewood) which had originally been a minimum of 3 inches deep. After 1 year it had basically 1 layer of intact chips/bark/large pieces on top (stuff that could dry out in the sun), and under that was 1.5 inches of what can only be described as wet sawdust compost.
It just doesn't hold up to being ground together every time I step on it, hour after hour, day after day all year, combined with being essentially wet all the time and then buried in snow for 5 months.