Hello Jim,
Definately not a good situation. I would fix it for sure. If it fails the car will be helplessly stranded. It may also cause some strange vibrations under acceleration.
I would install a used flywheel or get a good used flex plate and have it re-riveted to the flywheel.
In any case the replacement flywheel would have to be counter balanced as the original. They do this by taking the original flywheel and bolt it up to the new one (180 degrees rotated) and then balance the assembly doing all the adjusting on the replacement flywheel. In this way the replacement flywheel becomes an exact twin of the original and has the same counter balance.
You most likely would not have to do any balancing if you just changed the damaged flex plate. Not sure who may be able to re-rivet this for you, but it sounds like it would be worth looking into.
I should have some flywheels around here if you decide to go that route let me know.