IMO it really is just a matter of aesthetics in a general purpose ring.
Looking at it from a cost benefit the corners of a square ring are rarely used. So there is X amount of costly square feet paid for but offering little to nothing.
From a maintenance point as others have pointed out they don’t get dragged, “So they grow weeds”.
Installing rounded corner fencing doesn’t, shouldn’t cost more. It adds a bit more labor time but marginal for anybody that knows the drill.
I am a symmetry person also. Rounded corners are more pleasing to me.
Sidebar; funny how different areas use different names/terms for the same thing. In my neck of the woods which pretty much encompasses the mid-Atlantic I rarely hear “Arena” most people use “Ring”.
edit; forgot to add, I wonder how many rings started out with square corners and over time due to the “dragging” issue end up with rounded. Our ring came with the farm and was installed decades ago. Never “explored” the possibility it was constructed square. Hadn’t been used in a long time and was cover in grass/weeds when I purchased the property.
Square rings in a “pro” set up, used for shows also have a higher maintenance budget then most of us. In other words they have grunts on the payroll to do the BS manual labor needs.