I think you're stuck with the roots, I had a tree like that once and they were just all over the place. However on that issue and on the issue of clay soil, the real question is whether you can grow grass or not. Clay soil is a drag to deal with, grass eventually did take hold in my case but it took a few years to look like a lawn at all. I have seen on tv where they'll put organic matter back into the soil as you are probably contemplating. Not sure about sand; sand does not absorb water and if you have clay down 3 feet, it might turn out to be the same situation. Sand or granular matter in a houseplant's pot works because there's holes at the bottom and the water can run out, but does the rain water have any place to go ?
I think the french drain really does go by the foundation. That is SO much work though it may be worth paying someone to do it, or pay them for their advice. For example, does a french drain function the way you want it to in clay soil ? I have no idea. I cannot imagine shoveling a trench in clay soil but if your crawl is wet it's gotta be done.