Do not think eBay affects UPS that much; for one thing most eBay sellers are dealing with rather small packages, which cost dear using UPS versus the USPS. Indeed since USPS introduced flat rate Priority mail service, it is much cheaper to send smaller items that route.
As someone who has received damaged items via UPS, have to say most of the time it is due to improper and or poor packaging by sellers. Many eBay sellers are looking to make as much from a transaction as possible while minimising their costs. Now that the USPS has cracked down on them using their boxes, many sellers resort to any cheap box found in the rubbish, regardless of it's rated weight/crush proof limit. Packing materials (if any) are limited to a few scraps of yesterday's newspaper at worst, and at best some reused bubble wrap or peanuts.
Have always found UPS willing to pony up when damage can be proven to be their fault (such as an item arriving wet), but when a package is not properly done, they can and should refuse to compensate.
For the record, would NEVER ship any appliance via UPS unless one was willing to invest in crating and or a truly professional packing job. Have had two Thor GladIrons and a Hoover Twin Tub sent via UPS. All were eBay purchases and arrived intact. Well ok, the one of the wheels on the Hoover poked through the box and was damaged, but that wasn't UPS's fault.
One reason was so thrilled my Pfaff was fixed via an in home service call, was dreading crating and shipping the thing. The process would have called for a professional crating/packing job,and that would have cost money.
Have also found it best to insure the heck out of anything sent via UPS. Parcels with high insurance values are taken a bit better care of, IMHO. My UPS driver certianly takes note when picking up a parcel and the system indicates high dollar value insurance.
As for the orginal poster's problem. DO NOT throw away the package and any packing materials, and contact UPS ASAP to arrange a claim. Sometimes UPS will insist the shipper begin the claim process, but whatever you do, get the ball rolling as soon after delivery of damaged goods as possible. UPS will not process any claims for damage without the package and packing, however.
L.