Which browsers allow users to whitelist specific sites for access by a browser extension?

Or else what is the !#?!% technology hurdle preventing users from making their own lists?

Chrome has this functionality, other Chromium-based browsers most likely as well. But how many users would actually manage access lists for extensions?

I will test out Chrome - thanks. Yet I was hopeful there'd be other choices.

I believe more than one user would actually manage access lists for extensions but you have asked a really loaded question given that browser users are not being given the choice here.