England and Scotland seem to dominate the list of schools established centuries ago which still exist today. How and Why? Were there schools established everywhere and the British ones survived? Or is there any other factor which led the British to found more schools than their continental peers?

All lists of this sort are inherently problematic, for the very simple reason that there is never an actual continuity of records which might actually validate the claim. Rather, as I pointed out in this earlier response to a question about competing assertions to be "Britain's oldest pub", the claims are always based on some sort of unprovable assumption.

I'll repeat here the short addendum I made to that post, because it directly addresses your query about schools:

Much the same problem exists elsewhere. It is very common, for example, for British public schools to make claims to extreme antiquity – cathedral schools often claim very early foundation dates on the basis that it was one of the functions of cathedral clergy to educate their choristers; hence it’s assumed that a school must have existed on the spot from the date of the foundation of the cathedral. Wells Cathedral School, for example, claims to have been founded in 909 on this basis, but there is no documentation proving the existence of a school associated with the cathedral at anything like this early date (and, in addition, the school definitely closed for a period in the 1880s and was then refounded).In the case of another claim often encountered online – that of the Japanese construction company Kongo Gumi to have been founded in 578 CE – there is the same lack of both contemporary documentation and proof of continuity of existence. The claim made by this company is actually based on genealogical charts of unknown reliability that were drawn up in the 17th century, and which purport to trace the history of the controlling family back through 40 generations to the firm’s supposed founder.

In terms of the specifics of your query, I think the key issue here is that the schools making the claims are "public" (which, in British terminology, actually means "private") institutions, which charge large fees to the families of children who attend them. Making claims to great age adds to the cachet of attending such schools, and helps in a small way to explain and justify the costs.