We realize your lack of experience is hampering describing what has happened or could have happened.
You don't have a points distributor: It is an electronic Distributor.
When the Distributor is FULLY SEATED, meaning the flat part in the original distributor is fully against the engine block, the 'Flat' nib on the bottom of both distributor shafts, is supposed to slip into a slot in the top of the OIL PUMP gear shaft. The oil pump is what will pressurize and pump oil throughout the engine when it is running.
The GEAR you see on both distributors engages with the CAM SHAFT and when the motor is turned, either by hand or is started, the distributor turns.
So, IF the distributor is not FULLY SEATED, the engine may turn over and RUN, but NO OIL will be pumped through the engine, because the distributor shaft is not seated, and the bearings and other components will seize from no lubrication. This won't happen immediately, it can take several minutes or longer depending on type of oil in the motor and how it is pushed, but the whole time the ENGINE will be screaming, there is something wrong!
If the engine is now SEIZED, and you did not RUN the engine - you were unable to get it started - the shop did it.
FWIW, that aftermarket distributor is dangerous junk:
A-Team's HEI Distributor has everything you need, all under one cap. With a high-quality aluminum housing and hardened steel dual bushing construction, it is built to last.
swperformanceparts.com