While I don't have time for a detailed answer, let me just say that while "all-in-one" sounds like a good idea, it does not work in practice. It is very difficult to apply and get even coverage and avoid getting opaque. One of my friends calls it "The worst thing to happen to a can."

Short answer is no.

Correct way---strip the wood first.

Problem will be that pine, birch, and maple will all take the same stain differently. All three will stain with blotches due to the differences in the grain consistency.

To get the same color---or as close as can be , you will need to get to bare wood, apply a preconditioner---basically a one pound cut of shellac(very thin diluted shellac----or a comercially prepared preconditioner. Then you will need to use different dilutions of the same stain or different colors to get the same color on all three woods.

One of the problems with Polyshades is that it blotches as well. It also does not penetrate the wood much at all and can slough off, bubble, or blotch from water---rings from glasses/cups or spills).

The easiest solution might be to prime and paint the whole set.

One of the main causes of your wood looking blotchy is due to a poor sanding job. Pine no matter how well you sand almost always finishes crappy .Using a sealer or conditioner as some call it would help.
No of this matters unless your going to start from square one and strip the set.

There is a way ! The problem without stripping is it will be a darker table.
1. Throw away the polyshades.
2. You need to sand the varnish to accept new stain(actually toner).You can actually test. The higher the grit the less the toner will take.I'd start with 320.If too dark move to 400.if too light move 220 etc.Google oil toner wherever you're from.A gel stain will also work.Toner is a thick gel stain.Rub in and wipe excess.Cover with Minwax handrub poly.PS!!USE oil because if you don't like you can clean off with thinner within ATLEAST 1/2 hr.