You are 100% correct that you need to add it to each service. This is a consequence of an intentional decision (keys cannot be duplicated). Streamlining it would definitely be an improvement.
That being said, it's not a problem in the real-world because FIDO is so sparsely supported. Hopefully PassKey speeds things along.
The relevant WebAuthn standard actually supports keys regardless of whether they can be duplicated. Even "virtual", software-only keys are supported. It's up to each individual service whether they allow the user to enroll such keys.
I should have been more specific. Test how many clicks it takes to add a key to one of your "mainstream" accounts. We would hope that services which support FIDO eventually gravitate to a single UX language, that also reduces the time taken to register new keys.
That being said, it's not a problem in the real-world because FIDO is so sparsely supported. Hopefully PassKey speeds things along.