Engine popped back thru carb and died, no start issue

A sudden loss of power can make the coil fire at any random point causing the backfire, so I wouldn’t put a lot of stock in that having anything to do with the actual problem.

I would be looking at that distributor.

I had to come edit this.
No crank means no connection from the key, or no ground to the solenoid.

Try jumping the solenoid under the hood.
Cranks?
Then it’s probably either the ignition switch or neutral safety switch. If it has one.