I am generally in the "invest" side of things since taxes and insurance don't go away, so I'm paying for my housing, anyway. But I think ultimately it comes down to whatever makes you feel better, either having the house worries "settled" or almost certainly making lots more money invested.
*However*, this is not your real problem. Your problem is how to keep your spouse from spending your savings. Paying off the house might sounds like a safe way to do this, but how long is it going to take her to think of something so life-improving that a home equity loan would be called for?
Or maybe more generally the problem is misaligned money approaches.