Use gmail. AOL has always been proprietary and apparently doesn't let you use a 3rd party emailer (I'm always surprised that they even let you access the entire internet now, for that matter).
Set up a special account, just for this and don't turn on the 2-part auth (and note that using that probably blocks real people from sending you mail, since the 'authorize' msg gets deleted by the sender by habit -- I assume any such I get are actually either phishing attempts or link to a virus laden site and never use them anymore).
Then, make sure you have added that new, special email to the Kindle authorized email list on your account. the first msg you had sounded like you are skipping this step.
Then, test it by using gmail itself to email a book to your kindle email address. Once that works, then set up calibre to use your new address. You can then ignore that account forever, other than to go over now and then and move all sent messages to the trash (to keep your space used down).