If you have 1Password it makes software-based 2FA so easy that you'll barely notice it's on. Though I haven't been able to find a single US bank that supports software 2FA.
Does 1Pass support 2FA or otherwise handle it? I have been using 1Pass for years, but use a different app for my MFA.
You could also look at Bitwarden. I'm not affiliated, but it's an amazing password manager (desktop app, mobile apps, browser extensions). Even better, it's fully open source; you can even run your own server to keep out a third-party being your (encrypted) sync server. I've been using it for a couple of years now, and have never had a single problem. Free for personal use of almost everything… an upgrade to include the two-factor auth you're looking for is a very cheap $10/year.
Yep. If you have the desktop app installed, look for Two-Factor Authentication under Watchtower and it'll tell you all the sites that can use it. It's easy to set up too, it can scan a QR code on your screen. Some people believe that it's more secure to not mix your password manager with your 2FA though, since it's not "true 2FA".