If you had a broken screw in carb, it's possible the carb is damaged internally. IF all else fails that is listed below, take carb apart for a look see. You may need to buy a new carb.
Neighbor had a weedwacker that would hit and then miss. Turned out the flywheel was rusted and the rust would short out spark.
How healthy is your spark when it won't start?
I have a 13 hp honda that was a PITA to pull start. I had to have full choke, gas lever set in a particular position, pump pressure turned all the way to zero, open sprayer. Sometimes I would flood it. Tried to give it away to son, he didn't want it. Then I put an electric start on it. It sat for a year. When I hit the key, it fired right up, no problem. I'll keep it now.
I think if I buy a new genny, it will definately have an electric start and be a quiet RV genny with a 110v 30 amp.