Allow only 1 selection in a price set with multiple price fields
Yes, the price fields are independent, and the design limits you to putting all the membership types/flavours into one price field if you need to force only one.

Of course, you could write some javascript to do something like what you want, though that would be a bit of a hack.

A simple alternative is to create separate contribution/membership pages for each membership type - so the visitor would be presented with different links for each membership type before getting a form. That allows your form to be simpler and have additional membership-type-specific messaging and behaviour.