What Should Be Part of a Full Widget Tree Test?

I would say whatever works for you, because I'm not the kind of person that will say, okay, this is the only good way. Because I think wrong is to have no tests at all. And if you struggle with full widget tree testing, then I wouldn't force it and find to keep more sophisticated tests as widget tests. But I would just try to write some basic navigation flows or user flows to make sure that the user will be able to use that app as it's intended to be used. Prefer to do as is also as you can.