You don't want gravel, either. Weed and tree seeds, leaves, etc will fall onto it and be difficult if not impossible to remove, so then you will have weeds, little trees and gravel; a mess.

What you can do depends on how far off level the ground is, light levels, types of trees, etc etc, so Yardvaark's suggestion of photos really is essential to answer your questions.

Thank you for the comments and suggestions. Attached please find several photos taken today. The ground is slightly tapered down south to north, perhaps by 10 degrees at most.

Raising the soil level around trees, especially near the trunk, can kill them. How about a woodland garden with understory trees and native woodland plants.

You'll not be able to grow grass under many of those trees. I can't imagine that mowing around them all would be the least fun. Most should be sectioned off into islands, but there is no way for us to have complete understanding or for me to give you graphic feedback as the photos are all piecemeal and in the wrong format. The islands should be mulch at minimum, but would be much better in the long run to have groundcover. As pointed out (more than once, I think) using gravel/stone mulch, as well as being much more expensive, would end up being a disaster, and a royal pain to make corrections or alterations later. You would not raise soil around the trees other than just an inch or so to fill depressions, making the yard smoother. (It would be unnecessary in a mulched or groundcover area.) While you may have removed some of the lower limbs, you have not done enough. Some remain and some are starting anew.

A nice feature of many groundcovers is that they "eat" leaves so that there is no raking in the groundcover beds in the fall. I had a high enough groundcover to lawn ratio that I was able to blow leaves from the lawn into the groundcover in order to get rid of them. I never raked or hauled leaves away in the fall.

You can't add enough gravel or soil to flatten the slope enough to make a suitable badminton court without killing trees. If there are small holes, you can fill them to even out the ground a bit to make for easier mowing, but I wouldn't do more than that. It's clear that the ground is getting enough sun for a light layer of shade tolerant grass, but even if the ground was level, this type of grass isn't dense enough IME to stand up to the foot traffic that a sports area gets. If you want a badminton court, I would put it in a spot that is level and has enough sun to grow dense grass. This might require a low retaining wall with proper footings and the removal of more trees or siting elsewhere.

The area you showed in the photos will be most easily maintained for now with mowing I think, though as the trees fill in branches to take advantage of the new light and space created by tree removal, it will not continue to support grass, so you may want to plant a groundcover - depending on the type of trees, you may have issues growing many plants since shallow-rooted trees compete for moisture and nutrients with herbaceous plants and shrubs. In my area there are some energetically spreading ferns that make a lovely ground cover in this type of area.

At least if you are in the eastern half of North America, you will have weeds, particularly woody weeds (maples, honeysuckle, buckthorn, oaks, etc.) since this area always moves towards forest if left to its own. So to my mind, the woody plants seeding in is a far greater issue than removing leaves from gravel in the fall. Mowing will discourage woody plants from seeding in - that's how we returned our brushy fields back to fields and how we keep the woody plants from becoming reestablished both in the fields and along the sometimes shaded road verges. And you can't mow gravel, so weeds (woody and herbaceous) would need to be removed by hand if you spread gravel since even a weed whacker can throw gravel to a point of being dangerous. Gravel is a fine mulch if you live in a very dry area, but not good in areas with a respectable amount of rain IME.

"It's clear that the ground is getting enough sun for a light layer of shade tolerant grass..." It looks like freshly seeded grass to me. The kind that will lay over and die as soon as the heat sets in and the shade develops a little more thicky.

Thank you guys for your insights and helpful suggestions. Sounds like we should probably give up. It has been disaster after disaster so we are thinking of moving to another location (we are in the suburb of Cleveland) with far fewer trees. First came the Emerald Ash Borers that killed many trees, then deer during the two harsh winters began acquiring taste for many of pine trees that they never ate before, leaving behind ugly "top hat" type or simply dead. After removing many smaller and weaker trees, now fall leaves from remaining larger trees are more than ever, a major headache in the fall. I have made multiple enclosures to collect leaves in areas of small depression. It has turned out to be positive since they turn to rich dark dirt after about 2 years and leveled the areas. I have never mowed the area under trees, only using a trimmer to cut the tall ones down, twice a season. Neither have ever seed grass under the trees, somehow, grass and all kind of weeds just simply grow in the shady areas below the trees. I did not know more soil covering the tree root will kill the trees. Why is that? No trees have died when I composted leaves around them. I don't have a good utilization of my backyard at all except a deck for rest. It is indeed very nice having lots of trees behind and cooling off the house too. What is the ideal concentration of number of trees in a given area? Cutting down more trees does not make the area more usable due to the uneven nature of the ground (tree roots, stumps, weeds, stones, rocks, composted leaves). Concerning the leave eating ground cover, that is very interesting. Any details?

It is the other way around .... groundcover "eating" leaves. Fallen leaves settle down into the foliage of groundcover where over the course of a year, they decompose, turning to compost/humus.

Tree roots must have oxygen and live in a zone (depth) that suits their ability to acquire it. For the most part it is relatively shallow. Adding soil over the top of existing roots smothers them. Those roots die. If the ratio of them to healthy roots is too great, the tree dies. This is a massive "grey" area where there is no such thing as a formal study that rates the smother tolerance of each species. Each plant varies somewhat in what it can tolerate. Most trees will tolerate a little soil added over their roots, but not a lot.

"Cutting down more trees does not make the area more usable due to the
uneven nature of the ground (tree roots, stumps, weeds, stones, rocks,
composted leaves)."

What has worked for use has been to dig out the tree roots and stumps, pull the weeds, and move the rocks.

Beautiful, useful areas don't just happen- they are designed and cultivated. It doesn't happen overnight but over the course of a year you can make a huge improvement and over time accomplish anything.

Like they say "If it was easy everyone would do it."