Yes. I won't use U2F on services that don't allow multiple keys because of this. I have two keys in different places so it's easy to use.
Right now I do something similar, but I have to keep a list of which accounts I need to add to key 2, and the key is offsite, so I have a several week period during which an account only has one key associated with it.
Now, personally I wouldn't want the phishable Authenticator as fallback, but it's definitely better than SMS for example.
You will notice your key missing, then you can disable that key with your backup key. With only a password, it becomes a lot harder to notice someone stole your pw.
With 4 U2F keys, people who stole 1 of your U2F keys gain that one factor for only the services that you tied to that keys.