I’ve never heard any friend talk about embarrassment over how they look ‘down there’. But rather, it’s more than uncomfortable, it often hurts.
I have, and for many where this is the main reason, some women cover that by saying they are scared of the pain and discomfort.
This isn't to say the pain and discomfort isn't a valid reason to be worried about them. It is, and I think there is a lot that could be done to make them less painful/uncomfortable, if medicine didn't generally have the attitude of "shut up women, stop moaning, get on with it".
Eg, I think if a smear test only required a "digital vaginal examination" in the same way men get their prostate checked with a finger in the bum (ie not painful, just a bit embarrassing), take up would be higher than it currently is. However, there would still be a lot of younger women who wouldn't go because the embarrassment and worry about being judged alone would stop them.
Similarly, if the process was equally as painful but the swab was being taken from your throat, most women would go, in my opinion.
There are definitely a lot of younger women not going due to embarrassment/fear of judgement.