After you file

Follow-up thought: although there's no way to check this now, I have to wonder whether I could have e-filed successfully last year by putting  a zero in the AGI field.

I don't have a clever idea about why the IRS site says you didn't file last year when you did. It seems like if they really thought you had not filed you might have heard from them about it, but perhaps not. If you either received a refund or had your payment check cashed, then they definitely saw your return.