FWIW I think the occasional jam with a semi-auto matters less than the ability to keep shooting most of the time without any kind of manual or visual distraction.
Some will reply that they can work the bolt smoothly every time, just as quickly as a semi can cycle, and with no change in cheek-weld or hold. Well, if you can, good for you, but I suggest that for most of us, that isn't true.
Moreover, what a semi may give away in accuracy to a bolt gun, and not all of them do, and most don't give away much, it gains in its ability instantly to correct for a miss and turn the next round into a hit. The hardest part to get right when rabbiting with a rimfire is range estimation, which is more usually over-estimated than underestimated. The result is that the first round often goes high and a quick adjustment has to be made. Meanwhile the rabbit is probably moving. In this context I'd rather have to do nothing more than adjust my sight picture and press the trigger again.
But this is just one style. You can get very clinical and accept nothing but a head-shot, and sometimes it's simply a choice between keeping the landowner happy with volume or the butcher happy with clean carcases.
Anyway, I wouldn't write off a semi-auto for the job. A 10/22 with a trigger job would be my S/H choice, as there are a lot to choose from (though my best .22 semi was a Marlin 7000), but if buying new I'd go for a S&W 15-22, which I've found to be accurate, very quiet (the polymer dampens vibration and resonance) and comfortable (polymer contact areas don't chill like metal ones). It also has the virtue of a straight-line layout and a collapsible stock, making it particularly suitable for use wih NV.