I can send you a crap load of red oak acorns.
You really want big oaks dropping acorns and branches that close to your house?
Initially they put down a deep tap root and growth can be slow the first few years. Then they start to take off. A mature tree can put out roots well beyond its drip line. Potted trees that normally put down a deep taproot tend to be hard to transplant since they don't get to set that taproot first.
I think the tap root trees would be your best bet, preferably from seed, although I'm sure you don't want to wait that long. So any nut type tree basically...oak, black walnut, hardy pecan, shagbark or shellbark hickory, chestnut, etc. You mostly want to stay away from trees that really search out water...like elm, willow, silver maple, etc.