There is a semblance of truth to some of this, but I'm afraid anyone not especially familiar with the UK could come away from reading what you've written with a very distorted impression of reality.
I may be misreading some of your remarks — please forgive me if that's the case — but to respond to some of the points you make:
The list of UK Prime Ministers and opposition leaders, particularly in the earlier part of the twentieth century, does contain many wealthy 'establishment' figures, including some with aristocratic backgrounds, but the suggestion that almost all of them went to the same high school and have the same cultural background is a gross exaggeration.
Churchill was the son of a Lord and an American heiress, and was literally born in a palace. Thatcher was the daughter of a shopkeeper and a seamstress, and grew up in a home without a garden or a proper bathroom (the family's toilet was in a small outbuilding — not an uncommon situation for ordinary people at the time).
Attlee was the son of a wealthy lawyer, and he grew up in a house with a tennis court and several servants. Callaghan was the son of a coastguard sailor who died of a heart attack at a relatively young age, leaving his wife and children in poverty; as a teenager Callaghan was eligible to go to Oxford, but he couldn't afford to. He became Prime Minister regardless.
Cameron's father was a stockbroker and his mother was a magistrate. Major's father was a circus performer who made a living selling garden ornaments, and his mother was a dancer who worked in a local library; the family was nearly bankrupted in the 1950s and lived in a small flat in Brixton. Major never went to university, yet he still became Foreign Secretary (meaning he was in charge of the FCO — perhaps your friend's mistake was going to university at all, though I wouldn't worry about him; the future King and Queen are also both St. Andrew's graduates), and then Chancellor of the Exchequer and Prime Minister.
Eden's family were landed gentry. May's father was a village vicar; both of her parents died shortly after she graduated from university.
Wilson was the son of a chemist and a schoolteacher. Blair's father was given up for adoption as a baby and became part of the family of a shipyard worker and his wife; he grew up in a Glasgow tenement with an ambition to become a barrister, which he achieved. Blair's mother was the daughter of a butcher; she was born in the flat above the family shop.
One could go on; there is certainly privilege in the backgrounds of many British Prime Ministers, but there is also poverty and much else too.
There are popular private schools in the UK favoured by the wealthy, such as Eton, Harrow, Westminster etc., and many of the UK's political elites were educated at those schools, but it is not correct that almost all of them were. Many of the pupils at those schools do have wealthy parents, but not all of them are from the same background (I attended a British private school on a scholarship. Thatcher attended a local girls' grammar school on a scholarship). There are also expensive prep schools in the US favoured by the wealthy and privileged; the US and UK are not especially dissimilar in that regard. The main difference is that British schools tend to have longer histories, and admit a higher proportion of students on academic merit due to old bursaries and tax incentives.
I am struggling to understand your remarks about the number of people that must be won over to become an MP in parts of the UK, but what you are referring to sounds a lot like rotten boroughs, which were — contrary to what you state — abolished over a hundred years ago, by the 1832 Reform Act. If you are not referring to rotten boroughs, then what you are describing is just a marginal constituency where the vote is tightly contested between parties, and those exist in all democracies that use the first past the post system, including the US.
If anything, some would argue that the electoral situation is better in the UK because gerrymandering — which is acknowledged to be a widespread problem in the US — is for the most part impossible in Britain, since districting has largely been removed from political control and handed over to the independent Boundary Commissions. I take no stand on the wisdom or otherwise of the current approach to that issue in the UK, but similar solutions are widely advocated in the US in areas affected by gerrymandering. It is not uncommon for elections in the US to be decided on the basis of a small number of votes — the US and UK are, again, similar in that regard. So few people needed to be 'won over' to change the result of the 2000 presidential election in the US that the election was effectively decided by a court, and a closely divided court at that.
I agree with your observation that the electoral system in the UK can be 'hugely undemocratic', but that is a generic complaint about the first past the post system, which is also used to decide all major national elections in the US.
I can attest to the fact that Latin (if not full classics courses, which are typically the preserve of universities) is still commonly taught in British private schools, but I don't see much evidence that learning it will do a great deal for you career-wise anymore in the UK; Johnson is the first Prime Minister in sixty years to have a classics degree. Quod erat demonstrandum.
