A couple problems that I see:
Turning walk mowers around in the rough would lead to some light triangular scalping of the rough. The reel remains spinning while the operator tips the machine back and turns it, but to get the reel high enough to clear the rough would require the operator to push the handle down far enough to get the reel high enough, which would be about knee height. This is hard on your back after awhile so guys would just stop doing it and clipping the rough. Actually, after awhile it might look kinda cool as it would be a quasi fringe in a way. Or you could use turning boards, but that's only doable at places with a budget like Tony's.
Assuming a bent to blue transition, you would start to see bent creeping out into the rough, or if the greens edge creeps inward, tall bentgrass along the surround. Bentgrass at rough height is an awful grass.
I assume that several areas would have roll offs that would almost always result in a "collared" lie, where the ball is resting on the greens height grass but has nothing but rough behind it. Pretty much one of the harder shots in golf.
All that being said, it is simpler, and I don't particularly like those courses with like 4 different collar cuts surrounding the green, as that is excessive.