Though the original question was specific to sampling of the cervix, I would point out that a lot of factors combine together that make paps very uncomfortable for some and no big deal beyond annoying for others. And since the female reproductive anatomy has such a wide range of variances the discomfort level is just as wide.
A good collection for HPV/ PAP involves taking adequate samples using the cervical scraper, the broom, and the brush. Some will shortcut and only use the broom method but it ends up in more repeat visits when the lab flags an inadequate amount of cells received. All 3 samples involve multiple turns of the collection device and contributes to discomfort as some overall level. I would primarily see most women struggle with the brush since insertion directly into the cervix is required to collect the sample.
A lot is out of your control - where you are at in your cycle, hormonal variances or menopausal related atrophy making cervical tissue more susceptible to pain, cramping, bleeding and so forth. Your cervix reacts to objects being inserted into it, and cramping is one of the primary reactions.
But then a lot of doctors are also afraid to prepare a patient citing the need to reduce embarrassment and overall comfort which is sort of backwards in my view. Getting your butt all the way off the table, knees wide aparts and legs down to the side, pushing the vagina downwards toward the table end to relax the pelvic floor all contribute to a more relaxed anatomy even though it might be more embarrassing. I always took the approach to tell a patient that a tradeoff for awkwardness of the positioning and exposure required was a better chance of reduced discomfort during the sampling and that was always appreciated. The other thing is that you simply let a woman know that it's going to feel uncomfortable but preparing for that reduces the anxiety factor since mentally you expect a lot discomfort, you feel a bit, but not as bad as you expected, and the brain overall says oh well, ok, that was not so bad after all. Works for the internal and rectal as well.