It's fine if you do that for two or three services. But my password manager has around 600 entries. If I would need to set up all of them on both keys and always switch the active one with the one in the safe location for setting up a new account I would go crazy. Having both of them in the same place kind of defeats the purpose of having two keys, as you would lose them together.

So better have a synced password manager and unlock the password manager with a hardware key.