The closest I've come to "wrapping" the tail was putting the braided end up thru itself a couple of times, putting it in a tube sock (usually 1s that were worn out for wearing anymore ourselves, not new), cutting slits in the ends to that we could tie it to the braid and then taping with folded over duct tape (so that it didn't stick all the way around the sock/tail) below the tail bone.
I braid up the tail, run the end thru the braid at the bottom of tail bone, run the end thru the loop made and "snug it down", then if really long, run it thru the other side of the braid at the tail bone and back down and thru the loop created. If needed, then fold the extra length and hold while putting on the bag or sock (a bag fits loose and usually hangs below the end of the tail - I've found tails outside the bag).
I also like the "bag" that has 3 tubes you put the sectioned tail thru and then braid that.
Sometimes we braided it up, ran it back thru the braid at the top of the tail and back down thru the loop created and didn't do anything else. A lot of our ponies left it alone - not a lot of switching, no rubbing etc.
and I often braid it up and while driving them. Their tails didn't stay up after this drive - I'd used Cowboy Magic on their tails to get knots out w/o pulling hair and the next morning the rubber bands were gone and the braids partially undone and by evening the braids were gone. But their tails looked good, LOL.
As far as what we used - I just used Mane and Tail Shampoo and Conditioner. WE usually dilute the conditioner 50/50 with water and leave it in the tail. It's meant to be a leave in conditioner, but I found that when I used it undiluted and left it in - our ponies rubbed their bums/tails - often catching and breaking tail hairs or pulling the bag/sock off and doing quite a bit of damage with that. And for snarls (have lots of "witch's knots" right now - haven't groomed anyone properly or regularly this year) - have used olive oil, WD-40, Cowboy Magic and the same conditioner above sprayed in and rubbed thru the ends before working on snarls. I have been known to take a while but completely work out really bad snarls saving 90% of a pony's tail/mane that way.
If left braided and up too long in a sock or bag (more than 6-8 weeks), the braided hair will fuse into itself and become a huge mat that is very difficult to undo. We usually drop braided tails and leave down for a up to 24 hours 1x a month when doing constant braids/growing out a tail.