Check the kill wire to the coils.
It can only ever be open circuit or closed circuit to ground and must never ever see battery voltage or the chip inside the coils get fried and they are then trash.
Have a good look at your old engine.
Briggs generally use a short ground wire on the solenoid and a single long power wire going back to the main engine loom plug.
Solenoid don't open = mower won't run.
If you have hooked it up to the kill wire then it will only get ground when the engine is turned off and only get power when engine is turned on so solenoid will never open