He wants to play as a starter and he probably wants to be paid big $, correction - very big $. We know that AJ went within an inch of being traded to Cleveland last mid-season for a 2nd round pick and it would have been to play as a starting QB. I don't think he's anywhere near as talented as Jimmy Garoppolo but he probably sees himself in the same boat as a proven good backup who has never screwed up and waited for many years and not been given the opportunity to replace the Red Rifle in Cincinnati.
While I support moving on from Matt Moore as our backup QB and finding someone else, I think that bringing in AJ may be expensive and create a QB controversy that we don't need (and it would be premature to give up on Ryan Tannehill as our QB). #17 understands Gase's system and was progressing well before his injury and I expect that he will come back from his surgery with something to prove. He will be hungry. Competition is good but AJ would probably only want to sign to be a starter and he'd be expecting $12M+ (which is more than we paid Cutler) compared to the $2.5M or thereabouts that we have been paying Matt Moore - who is now a free agent.
I guess that it depends on the demand for AJ's services. If there are teams wanting his signature to be the starter, then he will be well paid. However, if teams are only offering him mid backup money, then that's all they see him as - ie backup insurance. In a way this is a bit like how the Chicago Bears signed Mike Glennon from Tampa last season and also drafted Mitchell Trubisky. Glennon (6'6" with the biggest neck in the NFL) was a long time solid backup and signed a 3 year $45M contract with $18.5M guaranteed. However, it's near certain that he will be cut this year because he looked underwhelming when given the start and not up to it, got replaced and he's definitely not worth the $15M a year. The Bears overpaid but they had a bet each way by drafting a QB and recruiting an experienced backup as a transition. Glennon will also be considered as a possible backup by Miami but probably for just $3 - 4M a year but the Chicago stint has shown his limitations.
I doubt that we've got the cap space to pay both Tannehill ($19M) and AJ McCarron both starting QB money.