There is the issue of not being able to get similar slick lock times or a trigger with the expected sensitivity.
Then, the underlever generally used can be a pain to operate around your bag & the alternative thumb lever heavy in function.
As was mentioned, with no primary extraction function, removing cases can be a pain. When it works, the usual straight back extraction can be a bit disconcerting while guide bumps designed to modify that can interfere with slick reloading.
Fit tolerances are more difficult to achieve in mortised alignments than can be turned with conventional actions.
They're cute little buggers, though, aren't they? I have a 1915 model Stevens Favourite rimfire & a fair used Annie barrel that I'm going to mate up some time & pretend that I'm out on the prairie......