Can't say I've ever heard that one. Any cogent reasons why not?

You can say both. To me, most well-established sounds slightly better than best-established. But there's very little in it.

I don't know about cogent, but a) someone once corrected my "most well-known" into a "best-known", so I assume they thought it was better, and b) you don't say "I feel more well", you say "I feel better" ...?

aber Komparativ geht doch nur mit better, oder? Nicht "more well known as..."?

Perhaps they liked "best-known" better because it seems a bit more 'positive' than "most well-known", which is fairly neutral. A phrase like "the best-known serial offender" seems somehow odd, but "most well-known serial offender" seems okay, to me at least. Or to turn it on its head, "the best-known family TV series" might be regarded as preferable to "the most well-known family TV series".

#5 That's true; can't remember if it was a positive "well-known" but it could have been that.