Not to mention that it makes me see all the emails I hadn't seen I had received since the last time I had opened Gmail 5 days ago.

that's the biggest issue by far and why email as 2FA should only be done under certain circumstances

Sorry I'm not sure what you mean by this. Isn't that an unrelated problem?

hm, desculpa but you failed to address the point. lose your key -> lose your account. Whereas with typical e-mail based accounts you forget your pw -> reset your pw