Still, I don't mean to knock the sling shot. Perhaps I should pick one up and seriously try to learn to shoot it well.
It's always good to at least handle what you're going to pooh-pooh before pooh-poohing it.

How many shots should one expect before changing the power band?
Depends. Usually I get several hundred shots. With some bands, even more. Flatbands don't last as long as tubulars, but I enjoy shooting the flatbands more. Tubulars for Dankung style slingshots last a long time. I've probably got a thousand shots on some of those bandsets, and they're still going (but these high shot count bandsets are lightweight target bands shooting light ammo, not heavy duty hunting bands shooting lead).
Keep the bands stored in dark containers away from sunlight and they will last far longer. UV really destroys rubber.