I have been in the same situation as you since 32 weeks when it started looking like baby was going to be big. This was confirmed by a scan at 36 and 3, where she was already weighing 7lb10 according to them... We don't yet know if this is accurate as I'm currently 38 and 2.
For me, the doctors did overplay the risk of shoulder dystocia, and I actually ended up semi-complaining about it (I was told it was a 50-50 risk, but it's far FAR less...). Also, I wasn't given the option of an induction, and the path I was being guided along was an elective c-section. In the end, like others have said, you'll never know what would have happened had you chosen another route, but for me, I just couldn't take the risk of her getting stuck. C-section was my LEAST preferred birth plan, and because of baby's size, it's become the plan.
I do feel for you, because once they get to that size it feels like there's not much that you can do about it! With our baby, my hunch is that she's smaller than they think but still quite big. I have very narrow hips and am getting pretty bad pelvic pain; I'm also not confident that her head is engaging as it's meant to - - last time she was measured as only 1/5 engaged. My bump, though not looking that big really, is so heavy I can barely turn over at night!
I don't know whether a c-section will prove to be the "best" option, and I'm scared about it, but all you can do is try to determine what you feel best about and then start thinking positively about it. In my antenatal group, at least half of the girls have ended up having emergency c-sections for various reasons, so this has calmed me a bit about the choice.
FWIW, I've been asked to take part in a Big Baby study, where the two options are elective c-section and 38 week induction. This tells me that even the doctors don't yet have enough data either way. Other than reading the evidence-based birth site, which is great, you just need to go with what you feel most okay with. I guess it's rare we get the births we hoped for, from what I've seen from friends and read here.
Good luck and report back - you're not alone in being in this conundrum! I think my baby's size is also due to being pretty much house-bound all year, though I've tried to exercise a bit. I wasn't overweight when we conceived but have put on more than I feel comfortable with. I wonder if Covid-era babies end up being statistically bigger...