We use our arena as a dry lot in winter with the good footing. Arena is actually set inside away from the fences about 10ft. We have railroad ties down one side to hold the footing in place with a wide grassy area about 50ft wide outside the railroad ties. We were doing actual hunter things when we built the arena, using mostly outside courses. Had unfenced fields for jumping at local shows. We did not want horses relying on fence to work off of, be dependent on fence to obey cues. It has worked very well in our opinion for the results we wanted. We do driving things now, arena works as well for driving as riding, especially the no fences part, to get wheels, pole or shafts to caught on.
So moving the riding area inside the paddock, away from fences will keep you safer, add better training on the horse as YOU signal turns or use your legs. They don’t get to make decisions during training! You have the wide margin, then real fence, if horse veers out of arena edge for some reason, to keep him in the paddock. You could use landscape timber edging, or keep the arena footing worked, let the grass grow for mowing, to define the edges. It can be comforting having “escape space” where horse is not rubbing fence with your leg! They all have stupid moments, are disobedient, slow to respond, so your open area comes in very handy. Visually, I think no tight fences aids in keeping things open, they feel good going bigger, faster if asked. We do a fair amount of extended work, they don’t self-slow at corners with no fence there to visually warn of turn.
I would not build our arena any different if we had to build another one. No fenced edges close by to get hurt on.