But contrast the situation of the United State of America, whose "Supreme Court" rather than an independent institution is in fact filled with partisan hacks to such an extent that it appears many on the Right felt that it was worth the obvious downsides of Donald Trump as President to get more Supreme Court justices.

Even the Law Lords (as they were typically called when the exact same body existed as notionally part of the House of Lords prior to becoming the UK's Supreme Court) were way more independent than that. A Government minister can say "No" to the pick of the committee which independently chooses candidates for this job, but they can't pick for themselves, the committee can just give them the same name again until (inevitably) the minister gets replaced with one who doesn't say "No".

It would in theory be easier for a British government to abolish their Supreme Court than for the US government, but in reality in both cases it would cause a constitutional crisis. The UK is quite capable of having a constitutional crisis, you don't need to write your constitution down to do that, but it isn't very likely.