I am one who has painted damn near every piece of furniture I own, so take my advise with a grain of salt
I would either paint it, or leave it be. MAYBE try touching up the scratches with a stain pen or something if you decide to keep it as is, but the whole process to re-finish wood for staining is incredibly tedious, and not something that I would want to do as a novice, and particularly not with a piece with so many details. You truly have to make sure to remove every bit of poly or varnish and sand it all evenly to get a uniform look.
I love the French Provencial/Hollywood Regency style of furniture and paint and some glaze could do such amazing things to that vanity! Not to mention, it's YOUR furniture. So do what you love. Everyone is going to have an opinion. My boyfriend and I bought oak cabinets for our kitchen off craigslist, which were very good quality, but so not my style. They were like 20+ years old, and an outdated very light color. We painted them white and they look amazing with the style of my kitchen (I have a 50s diner thing going on) Yet, I still hear b.s. from my Uncle for the sin of "painting oak".