In addition to feed complications due to the rims, rimfires can't handle rifle pressures. The .17 WSM is the highest pressure rimfire I know of, at about 33,000 psi a bit less than high pressure handgun cartridges, and about 30,000 psi less than 5.56. The high velocity .17 WSM is also about 500-600 fps less than the mild .17 Hornet, 1000 fps less than the .17 Remington Fireball and 1500 fps below the .17 Remington. Scaled up to 6 or 6.5 mm and velocity will drop below 2000 fps, go with a bigger case for capacity and you lose any advantage in cost/materials over centerfire, and the pressure difference means it will never be as efficient as a centerfire.
That strange high-performance rimfire rifle that George Kelgren was screwing with back in the 1980s that Olaf Larsson mentioned had pressures clear into the 60,000 PSI range, if I recall (I have a detailed article on it somewhere... somewhere). How, or indeed if this worked, I am not sure. Normally rimfire has a low pressure ceiling, like you say.