I work in obstetric theatres. Emergencies, but Electives are very similar.
Unfortunately it's impossible to keep your underwear on for a C-section. Not only is there the placement of the catheter, but there is also the cleaning of your bump before the drapes go up. There's also a "clean up" of your vagina afterwards to remove any accumulated clots and waters left behind. It's also to check there isn't further bleeding coming down out of the uterus. If you consent, most people also get a diclofenac suppository at the same time (have had 2 myself, do recommend).
We can put up a screen to stop you seeing the catheter going in, and the drapes are still up at the clean up and suppository stage. Any other time we keep you covered as best as possible. But for the cleaning, we need to completely expose everything below your breasts for a minute or two to allow for the cleaning, and for the solution we use to dry.
There's at least 5 people in theatre for the catheter, everyone (at least 7) for cleaning, and 6 during clean up. This isn't including you or your birth partner. Those that don't need to be down the bottom end try their best to not be down there, but will still have jobs to do inside the theatre, so can't really leave.
If it helps any, during my vaginal birth, I had minimal examinations. I can only remember 3-4 total, 2 of which were for the pessaries as I was being induced. I was covered throughout, unless moving around, until the time came for pushing, and the only people present then were the two midwives (required a second as had had diamorphine) and my two birth partners (pre-pandemic).
Have a chat with your midwife and consultant about it, and hopefully you'll be able to come up with a plan you're happy with. I'm always happy to answer questions.