The elevator button idea has come up before. If I remember correctly, the opposing argument was that anyone wanting to "grief" other players by "claiming" the bldg. in such a manner, would just make it easier for them to do so.
Since we can't test the "button" idea, I don't know if it would help or hinder. It's possible that "claiming" a bldg. has more negative benefits than positive. There has to be another way. I think it's manners and etiquette, but that doesn't seem to be working
The only way to solve this problem is the latter part of your suggestions list. Players just need more servers and more bldgs. And I agree that because there are less and less possible chances that you will find what is required to finish a task, it does seem to generate bad feelings and toons against toons. It was really nice when we had 30+ districts, so even though there's less now, I think Disney is working on it. At least I hope so.