I'll happily take that chance for the convenience. Even if 1password leaks, they don't have the keys to my vault. They would need my key and password to unlock it. The only time that isn't needed is if it's unlocked, which only is on my linux computer, which means they need to find an exploit with their app. In the 7 years I've used them I've never even heard a wiff of something even small happening.

That's still a single point of failure.

So is TLS or the compromise of a major root certificate authority, and those have no bearing on whether an approach qualifies as using 2FA.

The question is “How vulnerable is your authentication approach to attack?” If an approach is especially vulnerable, like using SMS or push notifications (where you tap to confirm vs receiving a code that you enter in the app) for 2FA, then it should be discouraged. So the question becomes “Is storing your TOTP secrets in your password manager an especially vulnerable approach to authentication?” I don’t believe it is, and further, I don’t believe it’s any more vulnerable than using a separate app on your mobile device (which is the generally recommended alternative).

What happens if someone finds an exploit that bypasses the login process entirely?

Then they get a copy of your encrypted vault. If your vault password is weak, they’ll be able to crack it and get access to everything. This is a great argument for making sure you have a good vault password, but there are a lot of great arguments for that.

Or do you mean that they get access to your logged in vault by compromising your device? That’s the most likely worst case scenario, and in such a scenario:

• all of your logged in accounts can be compromised by stealing your sessions
• even if you use a different app for your 2FA, those TOTP secrets and passkeys can be stolen - they have to be on a different device
• you’re also likely to be subject to a ransomware attack

In other words, your only accounts that are not vulnerable in this situation solely because their TOTP secret is on a different device are the ones you don’t use on that device in the first place. This is mostly relevant if your computer is compromised - if your phone is compromised, then it doesn’t matter that you use a separate password manager and authenticator app.

If you use an account on your computer, since it can be compromised without having the credentials on device, you might as well have the credentials on device. If you’re concerned about the device being compromised and want to protect an account that you don’t use on that device, then you can store the credentials in a different vault that isn’t stored on your device.

Even more common, though? MITM phishing attacks. If your password manager verifies the url, fills the password, and fills your TOTP, then that can help against those. Start using a different device and those protections fall away. If your vault has been compromised and your passwords are known to an attacker, but they don’t have your TOTP secrets, you’re at higher risk of erroneously entering them into a phishing site.

Either approach (same app vs different app) has trade-offs and both approaches are vulnerable to different sorts of attacks. It doesn’t make sense to say that one counts as 2FA but the other doesn’t. They’re differently resilient - that’s it. Consider your individual threat model and one may be a better option than the other.

That said, if you’re concerned about the resiliency of your 2FA approach, then look into using dedicated security keys. U2F / WebAuthn both give better phishing resistance than a browser extension filling a password or TOTP can, and having the private key inaccessible can help mitigate device compromise concerns.