It will be solid when installed. The only potential play you might get would be from the receiver.
Yeah, even when bolted in from the side, the bike racks still move around a little in my experience.
The quick release receiver block is snug and motion free though due to the wedge design I suppose. I'd expect some play to develop if used a lot for towing as the contact surfaces start to wear, but a bike rack is going to take forever to cause the same wear.
I've seen a multi-bolt stabilizer somewhere, maybe on Rovers North or Atlantic British, that would slip onto the receiver first, and then also overlap the mounted 2"x2" bike rack tube. It had bolts all around it so you could crank them all down from all sides which would immobilize the movement.
Seemed pretty cool but I never got one.