Ok, I resolved this.
I “duplicated” the form that was working, and put it on the page that had the form that wasn’t working.
Autofill did work there.
And now both pages work fine.
Still don’t know WHY this happened though. I created several new forms, but they all had the same problem. Duplicating seems to have done the trick however. I wonder why the form that I duplicated worked in the first place. (And it wasn’t because it was the first one that I created.)