You're right
But Carrick still is a nobody as a manager. Why'd we want a nobody in charge as a manager even if it's as a interim? You wouldn't want a random 3rd division player from Spain as your starting CB would you? There's more respected managers in Qatar alone.
Chelsea have had multiple half decent interims. It can be done.
I disagree that Carrick will have deserved it. Have you met our board? Not being utter shite, being an ex player and protecting the glazers can land you this gig.
There's nothing Carrick can do to deserve the job 'till the end of the season in a handful of games.