I personally am fine with this.
While you are adding this anyway consider using an open source app instead of google auth like aegis. There are many others but I wish I knew about them sooner.
I personally love keeweb. Passwords and 2fa all in one place.
I mean you could argue that defeats the purpose of having 2fa, but it’s convenient
It weakens it a bit, but in my opinion it still has strength where it counts. If an attacker gets access to your password outside your password manager (man-in-the-middle, keylogger, phishing), then you’re still protected. Maybe it’s hubris in my own ability to keep my password manager safe, but I’ve never been worried about storing MFA in my password manager.