In practice, the HoL isn't that bad, but it remains explicitly anti-democratic and it's a corruption magnet, since the Tories have started selling seats in exchange for donations. The value of the Lords is millions of pounds in Party donations per year.

The US counterpart isn't so much the Senate as the Supreme Court: permanently appointed partisan lawmakers. The US Supreme Court is limited in that it can only make law on the cases brought before it, but it's been a critical flashpoint for so much social change in the US. To a certain extent the US "gets" to have a dysfunctional senate because critical civil rights issues are fought in the courts instead.

(The people complaining about "judicial activism" have a very real point that a lot of critical US law comes from the Supreme Court, it's just that that includes basic things like "allowing people of different races to marry despite what Virginia wants")

It is also a problem that everyone runs to the US for comparison and not, say, the UK's elected devolved assemblies. Which seem to be reasonably functional with their proportional systems. (Stormont is dysfunctional because of the civil war, not structural issues in the assembly itself)