Yes a valve adjustment is in order since it turns over withOUT the plug in it......... That is common on these OHV engines all the time....
I do these quite often.... Then when a shop tells the person a new starter and other parts are needed...... Then they come to me for a 2nd opinion I adjust the valves and they saved a bunch of money in the meantime.....
If you need any info on how to do that job let us Know Mon Ami .....................