The references in the contract of carriage merely mean that the business arrangement between Sky West and United inolves an agreement to be bound by UA's contract of carriage.

So, legally, UA would be justified in disregarding this and pointing to SkyWest (although I'm not sure what the legal claim would be against United even if it were entirely on UA metal, since you were transported, just very late and very poorly served, neither of which is a legal matter).

However, having said all that, UA should obviously step up and mediate in service problems like this, because it's a copout to claim that your licensees have nothing to do with you. UA benefits from making UX look like a full-fledged part of the UA route network, and that requires maintaining that appearance in good times and bad. UA should go so far as to provide appropriate compensation on behalf of their licensee should it come to that. If this becomes a liability to UA, then they should either force UX to repay such compensations as part of their UA contract or they should fire that licensee and replace them with one that will generate fewer problems.

P.S. My understanding of direct has always been that it represents a single flight number with 0 or more intermediate stops and plane changes. Therefore, all non-stops are by definition direct (1 flight number with 0 stops), but not all direct flights are non-stop. Therefore, it is redundant, but possible to say "direct, non-stop"