Here's a method to consider: take the players who've featured for us; make teams based around all our formations that brought us success at the highest level (Europe); place said players in these systems; after you've made the first set of xi players, do so again with the second string of xi players. Whittle your way down until you find a spot for each midfielder. This will establish their place in terms of tier.
Carrick's competition is Crerand, Butt, Ince, Fletcher, Muhren, Moses, Stiles, etc imo. Putting him up against our most famous (or infamous) midfielders is just an exercise in besmirching him, as it's both unfair and way outside of his standing.
Amongst the aforementioned, however, there are arguments to be had for and against Carrick verses most of them. The problem Carrick has is that in terms of charisma, aura and projection, he's extremely atypical of a United midfielder as he has no bite and is not at his best in the biggest games whereas the likes of Ince, Crerand, Stiles and even Butt and Fletcher rose to the occasion and gave their best performances when called upon. On the other hand, Carrick is a solid, consistent performer throughout a league campaign and has a body of work in that respect that none of the others can match. If you rank him with that kind of approach, he has to shoot up in terms of regard because he was a component in league-winning or challenging sides for longer than any of the others.
Carrick's five best games in a United shirt aren't going to be universally agreed upon, I don't think, and he doesn't have a career-defining moment or extreme peak, which again, is quite atypical for a United midfielder. In terms of playing style he only has direct competition with Crerand as a DLP/defensive midfield hybrid (from the aforementioned grouping) , and of the two, Crerand's the one I'd pick first in a draft format. What does all this say? First and foremost that we've got a storied élan jostling for position in our midfield tree. It's very easy to misinterpret Carrick's standing as a knock on him rather than acknowledgement of how many good midfielders we've had - contextually, we are a league-winning and European Cup winning club over a number of decades, and with that territory comes a lot of good midfielders, so it doesn't have to be a knock on Carrick for him to not be mixing it at the top whilst still being a very good player with a solid CV behind him... it's just that, in such company, he's not close to being the only one with credentials.
Another way to look at it is, if you wanted a DLP at the base of midfield who can both defend and play out, Carrick can lay claim to being there or thereabouts our third best in the role... probably because that's not the way we've played the majority of our football, but that's beside the point.