it would be awesome for this to become popular enough to see teams of people race each other out of the solar system

how much of a propulsion system could you feasibly pack in a cubesat?

Very very little, but not nothing. I've seen a few tiny deployable solar sails and a few tiny electric motors in my brief research for my video. They seem mostly experimental, to test out theories and miniaturization.

Enough to maintain your CubeSat's orbit and do a bit of pointing, but you'd need something bigger to have the propellant and power to get out of the solar system

I frequently wonder if alien didn't send RNA to Earth to start life here. It seems like that could be the payload for anyone shooting satellites out to the universe.