I agree that the female connectors in the Seagate USM spec-ed enclosure are very recessed. You have try different SATA cables and power cable to find ones that don't have plastic surrounding the ends that fit. Luckily Seagate does make it easy to take the drive out, but like you, it's be fantastic to just use it outright.
In my situation, I got lucky and my SATA data cables plug in fine, but the power cable was a different story. Luckily I did have a SATA power cable splitter lying around and that I just nail filed the protruding plastic on the connector end until it plugged in.