But to the best of my recollection your kid didn’t get offered a large scholarship by St Andrews. And he ended up with plenty of good options in the US, with money not being a deciding factor for you.
There are plenty of full pay US students who aren’t likely to get into T20 schools. Few of those would be likely to get into Oxbridge either (unless they were exceptionally gifted academically but didn’t bother with any ECs). The question is how many of those potential applicants are sufficiently strong academically that they would get a large enough scholarship from St Andrews to make a decisive difference to the cost comparison?
The UK is cheaper than full pay at top US privates. And finding out early is nice. Its a good backup for a lot of US applicants. But as @blossom points out, all too often when a student goes there, that choice is made because there aren’t any attractive US options which are perceived to have comparable prestige, regardless of price. And that can backfire.
(FWIW I went to Cambridge. My S would have gone to Oxford if he’d got in. He filled in the rest of the slots on his UCAS list and liked having other UK admissions in fall of senior year as a backup in case all went wrong with his US applications. But when he ended up with desirable US options, the UK was never under serious consideration. Conversely our neighbors’ kid went to St Andrews precisely because of the lack of attractive US admissions, and its been pretty disastrous, with failed exams, an extra year, mental health challenges, etc.)