In reality, your sail still helps you get the boat up even though it's not sheeted and centered. Watch the sail of a capsized boat being righted next time. You'll see the boat slowly coming up until enough of the mast is exposed for the wind to get under the sail. As soon as the sail fills with air underneath, it lifts off the water and makes 'the wing'. The boat pops up almost immediately. This is why it's so important to get the bows of the boat pointed into the wind. It's also the reason that if you capsize on a very light air day, righting will be much more difficult.

If you were to center sheet the mainsail while the boat was capsized, it would give it a head start - but we know what happens when it comes up with the main sheeted...it rolls back over the other way because the bows were never straight into the breeze to begin with. However!, if you have the sea anchor attached as Bill suggested, this should keep the bows pointed into the wind throughout righting so that when the boat is pointy side up, it is still pointed straight into the wind.

I have not tried the sea anchor but it sounds like a neat idea. And yes, they're readily available - try West Marine.