When I have a broody hen, I usually leave almost everything alone. We haven't had a lot of trouble with our broodies, though, and some have. Mine have lost a few hatches, but mainly because the humans didn't realize they were brooding in that odd place and accidentally cut them off from it. The only thing I do is mark all the eggs under a hen once I'm sure she's setting. If I see new ones, I take them out. (I do this during her rest periods.)
In the incubator, last time I hatched in cartons. I've since bought plastic egg carriers for camping that I modified for the job. (2 12 sized slightly modified gives you 48 egg capacity.) I am a little nervous about them because I don't want the chicks trying to crawl around on smooth plastic. Anyway, I think when the eggs are upright in cartons like that, the chicks have to work harder to get out at the end and can rest longer inside the egg. It keeps the chick from pulling away too soon. Where, the time before, I lost chicks for the same reason you lost the duct tape chick, I didn't lose any in cartons for that reason. The only real issues I've had are with low humidity and fluctuating temps. That's not the carton so much as the time of year.
