We took my son just before he turned 2 on the family slides at TL and BB. He did great on them. Most of the CMs did insist that he was holding on to the handles before they'd push us off, so we mostly just had him hold on with our hand over his on the handle in the beginning since he wasn't good about keeping holding on. Since we had to be holding on too, they generally wouldn't allow us to go with an arm around him unless that hand was also holding a handle (which means you're not really supporting your kid), and a couple of the tubes (can't remember which slides) didn't have handles he could reach if he was sitting between our legs. So honestly, we'd get going, then just wrap an arm around him so if he did let go of the handle, he wouldn't be sliding all over on the tube.
We never felt that he was in danger of coming out of the tube or anything like that and he absolutely loved the slides. He was maybe 25 pounds and 35" at the time, so really small weight-wise, but it was never an issue. Just make sure everyone's holding on when you take off, then wrap an arm around him if it makes you feel better once you get going and he will be fine.