Beyond the general awkwardness* of saying "Modern Sporting Rifle," I would feel like I was speaking out of both sides of my mouth if I started using that term regularly.
While the description is factually accurate (after all, ARs are indeed one of the top choices in shooting sports), I usually try to steer people away from "sporting purpose" discussions. I don't feel comfortable hammering on people that the 2A isn't about hunting/sports while simultaneously stressing that my carbines are Modern Sporting Rifles.
I understand the utility of the phrase, and I don't object to other people using it, but it's not a term that I'll use myself any time soon.
*I just don't like using longer terms where shorter ones will suffice.
Modern Sporting Rifle = 6 syllables
MSR = 3 (though most people will have to ask you what an MSR is, defeating the purpose of using an acronym)
Firearm = 3
Rifle = 2
Carbine = 2