Many use at least some square rails at home, so do some smaller shows-round poles are simply more expensive. I never for a second thought they represented any kind of safety hazard over round poles, never seen any evidence of that.
I have tested that theory personally having been dumped into them a few times over the years. No stitches, that’s kind of a reach, it’s wood, not a metal sharp edge.
I have had a round pole drop under a horse and roll with them landing causing a stumble so it’s pick your poison if you want to jump.
Thinking back on one of the worst falls ever witnessed, it was a clumsy, distracted youngster with a Pro aboard over a cross rail. Horse went right thru stepping on the round rails, got scared and started to bolt, stepped on a still rolling round rail and flipped over. If they had been squared…probably ended up upside down anyway. That horse went to a Dressage barn.
Thinking back in when I showed Trail in breed shows and the long defunct West Coast AHSA Open circuit, we saw both round and square lumber. Step overs tended to be square to encourage a bigger step, back throughs round to point to even a small rub.