What I have noticed the past few years for any European flights I have booked is that the US airline is not the one in charge they are just partnering with the European airline. This allowed me to get my points etc. on the US airline while actually on a different airline’s flight. I did experiment and found no difference in price if I went directly to KLM, or Air Italia or Virgin etc. vs. booking from United or Delta.
Just thinking now that my daughter’s recent UK trip, although booked on United, was actually Virgin Atlantic. Flight was fine.