You can ALWAYS kill the greenery. Try using it a while without spraying or weeding. We also have a sand arena, 90ft x 300ft, which over time has gotten green on top. Mostly grass, a few weeds. I just keep it mowed, which lets the grass overcome most weeds. I don’t mow short, it is set at 5", just like when I mow the pastures.
We have found we LIKE having the grass covering, same reasons given by folks above. The sod part is firm footing, doesn’t get cut up no matter how hard the horses travel over it. NO DUST on windy days! Before there were clouds of dust when using arena in dry times, windy times. I use the arena as a sacrifice area, horses graze off the grass to keep it neatly trimmed. I don’t get ruts using the edges.
We drag or level the arena as needed, the covering grows back. It really is easy to keep up, though we are not worried about arena appearances to folks going by. Arena gets a lot of hard use by ridden and driven horses, lots of hooves going at speed, no change in turf, no holes or hoof prints marring the surfaces.
I would leave it go to grass, do whatever maintenance it needs to suit me, and enjoy it. Don’t waste your money on spraying, just mow it as needed for a nice look.