Small Engine Troubleshooting Problems? Ask a Mechanic Now.
If you remove the spark plug does it spin freely? What is the engine make and model? It can have a compression release that is not working. Have You ran a compression test? Too much compression can cause engine to be too hard for starter to turn the engine.
See it is spins with spark plug removed. Compression is tested with a compression gage, in the spark plug hole. Most auto supply's rent them. If compression is well above 100 psi, the compression release may be bad.
You can try to adjust the valves, set them to .004 inch, intake and exhaust. If no difference, check the compression.
You have an OHV, overhead valve engine, here is a video on adjusting valves on a Briggs OHV engine. Your's will adjust the same.
try backing the broken part out with a sharp pointed tool, like an ice pick. If the screw is too tight for that, it may have to be drilled through and an screw extractor used to get out the broken part.