The reason I take second-order logic seriously is that it lets me pin down a single mathematical referent that I'm comparing to the realities of space and time.

How can you say this after having read this thread?

If you believe in second-order model theory, then you believe in set theory. (However, by limiting it to second order over the natural numbers, without going on to third order, you are not obligated to believe in uncountable ordinals.)

ETA: It is very imprecise to compare second-order model theory and set theory like this. Already model theory is set theory, of course, albeit (potentially, not in practice) set theory without power sets. I should just leave the model theory out of it and say:

If you believe in second-order logic, then you believe in set theory. (However, ….)