Both of these are reasonable points, at least for the time being. But think of this as a long term idea. It's hard to project how people will behave in the future, as Bitcoin becomes more widespread and familiar. I'm pointing out a potential concern, and if it eventually becomes the case that hosted mining catches on, and clients demand proof-of-honesty from these services, then here's a solution ready to go.
That "if clients demand proof-of-honesty" is a very big IF that seems to go against the spirit of the original bitcoin paper (demanding an algorithmic answer, not a human one).
A related concern is that full nodes are disappearing as the cost of both running, and *especially* bringing online a new node, increases.
I made a proposal to address this issue that I called a
rolling root, but even though nobody found a flaw in it, it was unpopular because it would require changing the protocol in some significant ways. I still think it should be done FTW.