I use Microsoft’s Authenticator and so far have had no problems. I also use 1Password’s built in 2FA functionality which is a little redundant but it makes filling in 2FA code on websites and apps very easy.
Using tools like 1Password and Bitwarden to manage your OTP codes, while arguably better than not using OTP at all, is a bad idea.
If you password for $service is exposed in a dump then you're ok but if your password manager is exposed then both factors of credentials are exposed. Using a separate app is definitely more secure.
You can use Google Authenticator without being a Google user, but Authy is another pretty reputable app which might be worth a look.
I started using Microsoft Authenticator for work, since I have several O365 accounts that require it. Ended up really liking it and use it for all of my TOTP now, with Bitwarden as my password manager. Also have a yubikey, for other purposes.
If you password for $service is exposed in a dump then you're ok but if your password manager is exposed then both factors of credentials are exposed. Using a separate app is definitely more secure.
You can use Google Authenticator without being a Google user, but Authy is another pretty reputable app which might be worth a look.