I guess I'm in the minority here about UPS.
We have shipped with all 3, UPS, Fedex, and USPS, and still go back to UPS. I'd probably wager that we ship more than most here on AZ as well.
Out of all the cues and shafts we ship day in and day out, we have to make claims on about 3 to 6 a year. Sometimes it's 2 sometimes it's 8, but the average is pretty good.
All the carriers have the same issues. The long boxes get caught on the conveyor systems and create a log jam. Boxes pile up behind them until the load is high enough and the log jam bends or breaks, letting all the other boxes go through. We have found that using the large triangular boxes is the best. These boxes give enough room inside that the box can get bent pretty good without effecting the cue or shaft inside. We usually wrap a cue shaft in foam and bubble wrap. Not so much that it's tight inside the box though. We want it to have some room to let the box bend.
As for claims, we get paid on pretty much all of them. They all get denied up front and you have to complain or appeal, but they do get paid. They usually deny because of "insufficient padding", but when we explain how we package and why combined with the fact that the padding had nothing to do with the damage, they go ahead and pay the claim.
By the way, don't use any box offered by USPS. They are very light weight. Of the products sent to us that are damaged, a significant majority are shipped via USPS in those light weight square or triangular boxes.