Nope. The paint does a great job of sealing up the pores of the wood. Stain does its thing by enhancing the pores and making the figure of the wood more prominent, whereas paint covers up all of the wood's more distinctive characteristics and makes it look more homogenous, an undifferentiated material through and through.

In order to get decent results with stain, all the paint has to come off.

Sealing stains are very good for things like chairs, or you can come back with a finish coat on top of your stain if you like, whether it was a sealing stain or not. You can paint a pattern into the back and then stain on top of that, but try it first on some scrap to test the appearance, you might want to mask the pattern when it's time to stain, and then finish coat over everything so it all has a uniform sheen. That will look very good, and wear uniformly too.

Good luck with it.

I have these really ugly chairs that I would like to refurnish (I love projects) and I think I would like to stain them, because I'd rather not have a solid color. I would also like to paint a pattern on the back and seat of the chair, like flowers or something. Should I get stain without a sealant and then paint over it before sealing it, or should I paint the pattern on and then stain over it with a sealant stain?