Arm drags are most common in 'pocket' range while hand fighting, which also happens to be face punching range, so you won't really see the same jockeying for grips in mma that you see in grappling, which includes arm drags. Same reason you don't really see 'wing chun' style hand trapping either (you'll occasionally see boxing style hand trapping). 'Infighting' in mma basically equilibriates into either, not infighting, or deep clinches (the cage makes an exception since it more or less acts as a 'third limb' in providing control, opening up the use of different techniques; kinda like ground fighting, but vertical).
If all parties involved are on guard and battle already joined arm drags have slim applicability. There is however, one application very suitable for arm drags outside of both the ring and the mat. That is, as a sneak attack.
Eight times out of ten the sucker punch wins the fight; immediate preemptive escalation to percussive persuasion as a first resort however, is something of an asymptotic proposition, that can carry some thorny logistical, psychological, and legal issues. A surprise arm drag to a go behind however, provides a pathway for a workman, officer, or john q. public, to put hands on someone and proactively defuse possible issues, without having to resort to a nuclear option.
You can apply the same principle with other methods as well:
A bicep tie normally is one of the least advantageous entanglements in grappling, in terms of both offensive and defensive options (not the least of which being you basically give the other guy an elbow tie); it is however something you can go 'directly' to, like the arm drag, not requiring the other person to be engaging with you at the same time, and against someone untrained or barely trained, the formal structure is less important than the substantial balancing.
Inside ties are also 'smothering'-
-in the sense that if the other guy wants to punch you or pull something out of his pocket he'll have to strip it first, and that pushback is what gives you the opening to go behind, where you want to be to control the individual with a full nelson or put him down with a mat return.
In many ways, you could see it as simply an extension of the 'fence' concept, of keeping the other person out of your personal space with arm extension and pushing, keeping them between you and them.