Hi Taylor,
Using a living tree should be fine as long as it stays living... Using a dead tree I consider a poor idea, as the roots will indeed rot in time. Can't say how much time...
Is it really faster to adjust the roof in order to deal with a tree sticking up from the roof, than to make a post from it and dig a hole for the post? How deep are you digging these holes?
If you wanted a front overhang, the tree could extend through this area, without the roof necessarily being leak-free around said tree... A tree that can be coppiced/pollarded would be good, as you could keep the upper growth to multiple trunk in manageable sizes, rather than a single huge trunk which could wipe out your wofati, and occupants, if it should fall. Deciduous would also be good, for max sun in winter and shade in summer.
Just, IMO, don't use a dead tree, or kill the tree in the process.
Another thread related to this here, including links to fancy tree attachment devices:
https://permies.com/t/45176/timber/Standing-trees-building-supports#359594