That is because no-one in the US could possibly conceive of a system in which college wasn't the thing that separates people. But if you are a Brit, you understand that almost all of the people you name went to the same high school and have the same background culturally...it is like all the Presidents you name not only going to the same colleges but also going to Philips-Exeter or Groton and having grown up in the North-East with parents who largely did the same jobs...that kind of thing is basically inconceivable to people from the US (the reason for this is a subtlety in how politicians are selected that, again, people unfamiliar with the detail of UK politics don't understand...to become an MP in some parts of the UK, particularly in the past, you only needed to win over at most 50 to 100 people...that is it, I am not talking about hundreds of years ago, I am talking about a decade ago...this changed slightly as both parties brought in changes to selection processes, but it is still hugely undemocratic in a way that would make no sense to anyone from the US).
Also, as I was careful to mention, the issue isn't necessarily that PMs went to the same university (although they did, your reasoning is backwards about Oxbridge...the UK has a large number of other very good universities that are better than Oxbridge in some subjects, the perception that it is a cut above is because all the PMs went there not because it actually is). It is cultural, and it is throughout govt...so in the US, you still see the same elite representation in the civil service...but it is nothing like the UK where there is close to 100% representation (the FCO is notorious for this, you need the right university, right high school, right primary education, and right parents...I know a guy who went to a £30k/year private school, parents went to Oxbridge, he got a first in IR from St. Andrew's a top university for IR, then got a distinction at KCL for IR again a top university globally for the subject, 4 high A grades at A-level...didn't even get an interview with the FCO). This is often very subtle because you have private schools teaching subjects like Classics...these subjects aren't really taught at all outside private schools, so if you go to a private school you can get into Oxbridge even if you are a relatively mediocre student if you apply to this subject (Theology is another, Land Economy is another...most students in the UK have no idea these routes into Oxbridge exist).
Also, as I was careful to mention, the issue isn't necessarily that PMs went to the same university (although they did, your reasoning is backwards about Oxbridge...the UK has a large number of other very good universities that are better than Oxbridge in some subjects, the perception that it is a cut above is because all the PMs went there not because it actually is). It is cultural, and it is throughout govt...so in the US, you still see the same elite representation in the civil service...but it is nothing like the UK where there is close to 100% representation (the FCO is notorious for this, you need the right university, right high school, right primary education, and right parents...I know a guy who went to a £30k/year private school, parents went to Oxbridge, he got a first in IR from St. Andrew's a top university for IR, then got a distinction at KCL for IR again a top university globally for the subject, 4 high A grades at A-level...didn't even get an interview with the FCO). This is often very subtle because you have private schools teaching subjects like Classics...these subjects aren't really taught at all outside private schools, so if you go to a private school you can get into Oxbridge even if you are a relatively mediocre student if you apply to this subject (Theology is another, Land Economy is another...most students in the UK have no idea these routes into Oxbridge exist).