I think there's a conflation of two issues here and that needs to be separated first and foremost. You are oscillating between defensive actions and contributions and offensive ones and they are two disparate issues when talking about any version of Martinez. I don't think there's any pushback whatsoever from anyone - even in his current state - that he's comfortably the best at the club at coming out of defence and doing something constructive with the ball. It's precisely why the Big 2 in Spain will look at him and why he'd very likely look like he's back to his best at either club.
If our onus is on consistent contribution from our backline to the offensive actions of the team, Martinez is the first name on the teamsheet from the CB's we have, so much so that you'd work out how to protect him and extract the best of what he is out of him even in this seemingly diminished state. If you gave me the brief of setting up the team so that we get the best out of Martinez, he'd go into the middle of the 3 and he'd have Ugarte and Casemiro ahead of him. We'd be playing a variant of what teams in the 70's and 80's used to get the best out of their sweeper, essentially. The contingency would be for others to cater to every whim and mistake Martinez makes defensively whilst enabling him to express himself optimally on the offensive end, but that's something reserved for very special players, perhaps the last of which was Sammer in a back 3. Is Martinez good enough for that kind of TLC? Amorim doesn't seem to think so at all thus far, but who knows whether he might try for it further down the line?
The problem here is the ratio between our backline defending and scrambling vis-à-vis being some masterful 3rd wave contributing to the constructive build up of the attack is heavily tilted i.e. you're far, far more likely to see the unit in a panic than you are to see concerted control, and then you're looking at these players from the defensive perspective first and foremost (which might be a d'uh moment, but still important to mention) and where we're looking at each CB for their individual qualities and foibles and why such and such can't/doesn't go here or there in the 3. Martinez is left-footed, which is the primary reason why he lines up at LCB; it's a no-brainer on paper, but opens him (and us) to the things he isn't good at as an isolated player. His lack of pace is important because he's going to be forced into 1on1's, which require that athleticism he lacks allied to the reading of the game to either nip that in the bud or use as a recovery asset, which he struggles with. As others have said, he's in do or die's a lot because of that - he knows he has to get everything spot on and that's a lot of pressure in itself; those with the recovery pace and athleticism can make mistakes and rectify them in the same play to the point people don't fixate on what they did wrong and only the end result that they still achieved the objective of stopping the man/ball. Martinez doesn't get that grace and he's perceived harshly more off the things that go wrong for him than the things he pulls off.
The problems for Martinez don't really end with a position change to the CCB unless we exert a lot more control on games as a whole, but then, I don't think we have this discussion if we are a controlled team - we cause our CB's to commit to things a lot more than should have to and I think that's part of the problem. It's like a set of keepers - all can look great and then a mistake or two can tarnish a whole performance - and I don't think Martinez gets put through the mill so much at other clubs that are front end.
BTW, I don't disagree with what you're saying, but you have to examine how and why that comes about and what needs to be put in place for it to be viable, which is where I think you see Amorim veer away from considering it. De Ligt I'm not sold on and Maguire is so so, but I get why they are put there over Martinez in our current predicament; us facilitating Martinez at CCB and not compromising even more of the team than we currently do feels like a remote likelihood to me. At least with De Ligt or Maguire you get the aerial presence and physical strength - they're both faster than him (eek) and he's not even much better than them over short distances. The reading of play isn't resoundingly better from any one from the 3, so you can't even say Martinez has that over them, where you usually expect the short CB to be the guy who is head and shoulders over others in this respect because he has to be to get by. I wouldn't mind us trying it; we lose games handily anyway, so what harm can it do to examine? You might be on to something, even...