Wood decomposes and grows mold, it causes allergy attacks and in general is super gross.
The rubber is one of those things that gets on everything. You leave the playground with black debris all over you.
In child care IF the highest point is 4' or less then we can use only grass. So if the highest point they'll be climbing on is short you don't need a high cost ground cover.
Pea gravel is a good choice because it absorbs the fall and it has no debris that gets everywhere.
We had a huge sandbox with our climbing equipment in it along with several other structures. In a neighborhood of tons of cats and we didn't have much of a problem with them using it. I had a staff person go out before we got the kids up from nap each day and they hardly ever had to pick up even a piece of trash. So just because it's sand does not mean the cat with use it.
I have a play set at my home that is very tall. The lower platform for the slide is level with my chin. We have always used grass and now that we've moved it we're planning on planting a more dense grass under it. I am not able to spend a lot of money on anything like you're talking about.
BUT if I was going to do a ground cover I would do gravel. 100%.