That broken flex plate is almost identical to one that came apart on me in a '64 Impala with 283 I had just done a re-ring on. I was 19 at the time and didn't know to check the flex plate when I re-installed the engine, so I convinced myself the thing already had stress cracks and came apart during a rubber burning start. I replaced it and the harmonic balancer (somebody suggested it, so I did it) and had no more issues with it. I bought a parts truck with a 4-bolt 350 in it a few years ago, and again, the flex plate had shed its center part, sheared off the torque converter bolts and destroyed the transmission housing.
Flush and check the engine, to include dropping the pan and cleaning out the settlings that's bound to be in there as others has suggested. Maybe you are lucky and will still have a good engine.