Matching tire requirement- fact or fiction?

I’ve wondered how valid this concern is, after my fifth friend had to buy 4 new tires instead of replacing the single non-repairable flat.

I recently went to a different tire store and they call b.s. on the matching tires thing, and I’m inclined to agree. At least, putting aside braking and such, like mixing summer tires on the rear with winter tires on the front, I’m just talking about issues with the differential, traction control magic, or who knows what else.

Isn’t the whole point of a differential to allow the wheels to turn at different speeds? And when your fronts are worn differently than the rears due to infrequent rotation, that results in the same condition of \*slightly\* mismatched tire circumference.

What gives?