Every burger has to go on something and traditionally that means a bun. Bread is out simply because it gets soggy and you can’t handle the burger anymore, and there is no way to eat a burger other than picking it up! Yes I’ve seen people eat a burger with a knife and fork but we all know that’s just deviant behaviour.
For the ultimate burger experience I like a bun that fits the patty nicely, isn’t too soft as to squish up and doesn’t get soggy and has sesame seeds (I love the toasted taste they add). It needs to be toasted, either in a dry pan or in a toaster on the bagel setting so only the inside of the bun gets slightly crunchy. Old time diners get the toasting beautifully. I’ve tried grilling the buns but it’s always an inconsistent result.
I’ve baked my own buns with a mixture of white and whole wheat flour and found that to be a good consistency as long as they didn’t get too high: for me the meat is the star of the show and the bun a supporting player.
That said to have a very nice burger you don’t have to be limited to a stock hamburger bun. Different types of pita work exceptionally well, as do buns like chiabatta or potato. Avoid anything sweet like egg buns or croissants.