There is a big difference between "won't work" and "won't work great".
Agreed. There is also a difference between "works great" and "works" (well enough).
Many designs over the years have proven that rimmed rounds can work well enough. They DO need a little more careful design work on actions and magazines, and you have to pay a little more attention to what you are doing when loading magazines, over rimless rounds, but other than that, they work fine.
.22LR should illustrate this point well enough, but if not, there is also the MILLIONS of .303 British and 7.62x54R rifles that use box magazines and rimmed rounds.
Also the "semi-rimmed" rounds like .25ACP, .32ACP, & .38ACP/.38 Super do have a small rim.
The .357 & .44Magnum rounds have also proven to work in box/stick magazines, in semi auto pistols.
The case rim, by itself, is not a serious drawback. There are still times in this world when the old simple systems are still effective and efficient.