The current state-supported model for UK universities is not sustainable. There are only a few British universities with endowments, and these and others that develop a long term plan for financial self-sufficiency are the only ones whose long term survival as third level institutions I would be willing to bet on.
I'm not sure where you're getting the idea that there's a lack of specialisation in US undergrad degrees?
Or the idea that 'independent analytical skills' are a thing, or that analytical skills (if that's what you meant) are not developed in leading US universities.
Progressing to a postgrad degree isn't always necessary in the US, where graduates of reasonably well-regarded universities can make very good starting salaries straight out of university thanks to the range of skills learned at undergrad level. Adding to the debt burden you may already be carrying isn't always a good idea if you manage to establish yourself on a career track, or even get a decent job, right out of university. One of my DDs might yet go back to school to get a masters in finance, but she's making a nice salary where she is so there's no pressure. Another considered an MBA but again, she's doing well without it.
Professional level postgrad degrees remain as popular as ever - law, medicine, pharmacy, veterinary medicine, dentistry - and there are always people going on to do an MBA regardless of academic background, because they have enough mathematics/ comp science/ stats courses under their belts to facilitate this. Applicants to law schools come from a huge variety of academic backgrounds, but there's nothing to stop liberal arts and sciences grads who focus on the arts or humanities from deciding to pursue something like medicine either. One of DD3's friends has a degree in philosophy and is currently in med school with hopes of becoming an OB/Gyn eventually.
Obv someone doing a masters in something like psychology or a L/MCSW qualification would probably have done psych at undergrad level, or at least enough psych courses to provide a solid foundation. There are masters level courses in areas like public health that attract applicants from a wide variety of undergrad majors. Same goes for library and archival science at masters level, human resources, etc. A masters in education will get you into teaching or high school counseling if you decided not to do a Bachelors in education/ counseling. Masters programmes that demonstrably lead to career tracks are not lacking for applicants. The cost benefit calculation comes down squarely on the benefit side here.