Air, fuel, spark and compression.
1.Remove fuel line AT carb. Drain into a glass jar. Should have a solid gravity flow. If not loosen gas cap and recheck. Might be a good idea to dump and clean the fuel tank.
2.Pressure test carburetor needle and seat. Video will explain it.
3.Take an old spark plug. Gap to 1/4" wide. Ground on engine block. Should have a nice blue spark. Don't forget to zip tie the dead man lever so it will spark. Spark yes or no?
Not related to a no start but.....
Remove metal engine shroud where starter rope is. CLEAN all under the flywheel, engine block and cooling fins. Look behind the muffler and around the oil sump breather plate. Remove all grit, grass and oil. A dirty oil layer will insulate the block and fins causing over heating. If you neglect this, engine damage will occur. THIS is a YEARLY maintenance item in ALL mower/small engine manuals.
Your engine has an air vane that operates the choke. There is a small adjustment on the muffler thermostatic arm. You center the silver muffler lever between the two marks. Air vane should move freely. When you move the vane, the choke should open and close.
VERIFY your choke is closed on a cold engine and OPENS when it warms. Remove the air filter and check to see if the choke plate is sticking CLOSED. If sticking tap on the choke plate with a small screwdriver. Should free it up. And if it is sticking, verify what is causing it to stick and fix it.
These are simple engines. Don't give up. We are here to help.
slomo