Gary Sanchez will be the Yankee starting catcher next season. Sanchez was arguably the best catcher in the American League last season despite playing only 53 games. However, Brian McCann wasn't too bad, hitting .242/.335/.413 despite a .269 BABIP. He was 9th in fWAR among AL catchers. Assuming the Yankees don't trade him, McCann seems set to catch 50ish games next season and serve as their primary DH against right-handed pitching.
Brian McCann would be the best backup catcher in the AL, and not just because he's a good catcher who is an above-average hitter. Backup catchers are dead weight on the roster. Teams have to devote one bench spot to a player who probably can't run, field other positions, or hit very well. Austin Romine adds no value to the team other than the ability to occasionally play the toughest defensive position in the game. McCann can cover that position while also providing value as a DH and pinch hitter.
I'm there the Yankees will hesitate to ditch Austin Romine. He's out of options, and they probably can't get much for him on the trade market. On days where McCann serves as the designated hitter, they'll risk losing the DH if he has to come into the game. Backup catchers often sit on the bench deep into extra inning games just to serve as insurance policies in case the starting catcher gets hurt. McCann as DH would make that move a little more costly.
That said, they should still do it. Using your DH as your backup catcher effectively adds a player to the bench. With Romine, the Yankee bench projects to look something like this: Romine, Torreyes, Austin, Hicks. Ditching Romine allows them to add a useful bench option for the late innings. Ruben Sierra comes to mind as a good Yankee bench bat to emulate.
The situation gets even more interesting if baseball agrees to expand rosters to 26-men. Now, the Yankees can consider a five-man bench with no backup catcher. They can carry two real specialists: maybe a Jarrod Dyson type pinch runner to compliment a Billy Butler or Steve Pearce-like lefty-masher.
A few years ago, I might have argued that the Yankees had little to benefit from a expansive bench. Players like Robinson Cano, Alex Rodriguez, Mark Teixeira, and Derek Jeter weren't going to be lifted in the late innings in their primes. However, the current Yankee roster looks primed for this kind of specialization. Gardner, Ellsbury, Headley, Gregorius, Castro and whatever happens in right field are all very pinch-hittable in certain situations. The Indians rode strong roster flexibility all the way to the World Series this season.