“pulling the mucus plug” isn’t really a thing. it’s a gob of cervical mucus that usually falls out before or during labor as your cervix dilates, not something you would pull.
I think the procedure you’re referring to is sweeping, or stripping, the membranes. which IS painful and can stimulate labor (but not always). it’s a fairly common procedure you can ask your OB or midwife to do for you if you’re overdue but not all will do it. it’s separating the amniotic sac from the cervix, not breaking the water.
I look forward to this piece! I had someone sweep my membranes without warning or consent when I came into the NYC hospital where I delivered, too, and it was really painful and violating.