So a few minutes after I made my second post, I tried to put the mirror back on the anchor button and the button promptly came off the windshield! I went back to O'Reilly, where I had gotten the adhesive kit and the guy behind the counter was very nice about it and gave me another kit at no charge. I went through the whole process again, cleaned everything off and re-attached the anchor button. Even though the instructions only call for 30 minutes curing time, I'm letting it cure a full 24 hours before I try to put the mirror back on. I've done at least 10 mirrors like this and I've never had so much trouble one mirror on.
I was reading online that Ultra Grey silicone can be successfully used for this purpose. Prep the windshield and anchor button as you would for a regular adhesive kit, then put a dot of Ultra Grey on the anchor button, mash it onto the windshield and hold it in place for a couple of minutes, then tape the button down with painter's tape for about 24 hours before hanging the mirror on it. If this second attempt with the adhesive kit fails, the third time with Ultra Grey may be the charm.