This is all too much like British exceptionalism.
It's apparently "irrelevant" that we still have aristocrats sitting in the House of Lords simply due to their ancestors being gifted title and land by royalty centuries ago. Also that the Church of England appoints Bishops to the House of Lords. Also that all other Lords are appointees of the government (albeit with a gentleman's agreement to allow opposition parties to also nominate their own Lords).
Also apparently "irrelevant" that the apex of aristocracy, the royal family, sits atop the parliamentary system while still owning huge wealth and land.
Finally it's apparently not a problem that there is no written constitution, despite the fact that we're just about the only so-called democracy not to have one. Instead we have a piecemeal hodge-lodge of legislation, coupled with gentlemen's agreements or "conventions" which the current PM demonstrated recently his government was willing to ignore when it suited its ends.
Without a written constitution, there's nothing stopping a party with a majority at Westminster from repealing the Human Rights Act, or the Scotland Act, etc.
As for the US, whatever your opinion of politics over there, under the federal system individual states, and also cities and counties, have far-ranging powers and hold public votes on a range of issues. The UK in comparison is a highly centralised unitary state, with a somewhat eccentric hodge-lodge of devolution, with powers very much In the gift of the UK government. Our local authorities are similarly relatively powerless, deliberately made so by successive governments.
So as I said, the UK is not really in a position to get on its high horse about democracy. Indeed, the British state operates more like an ancient oligarchy that has begrudgingly made concessions to democracy in order to protect its privileges and position.