I think going to one of these schools offers advantages and disadvantages. In general, British society is anti-elitist and even saying the name Eton creates many negative, if uninformed, reactions and views. An expectation is made on OE that they should succeed in everything and if they ever come across as arrogant or unfriendly OE it is because they went to Eton and not merely because that is their personality.
On the whole, Etonians are affable and friendly as they are aware that aquaintances can be defensive because of their schooling and they wish to dispell and myth. However in countries such as America and China, for example, being an Etonian or Harrovian is very well thought off and sought after. For those who have no negative bias towards the school it does have an advantage in this country too because they can see the benefits of employing someone who had such an education.
If you want to get DC into Oxbridge or Ivy League then Westminster is the school to go to as it sends about 60% of its cohort there every year. I have often wondered if Oxbridge looks better on your CV or Eton. I think that on the whole Oxbridge looks better if you have gone to any other school but I know very few OE's who have not had an interview because for some reason everyone wants to meet one!
I know plenty of people who have Oxbridge on their CV who have not had interviews.
Dependent on the employers some public schools are known for being "thick and rich", "parrochial and feudal" and "too hippy" although none that you have mentioned fit into this category.
In the City though, Harrovians have a reputation for being arrogant and scheming but again that may be because of their personality and not the fact that they went to Harrow. Unfortunately, my experience of Harrow boys has been a negative one.