I have drilled out several of those and made new ones on the lathe. I typically use epoxy to isolate the two metals when installing them and haven't had any issues.
That unit should have a pull stater with a rope if I remember correctly. The Bolt and washer are outside the outer cover and should be tightened to 36 ft Lbs as per pages 4-64 in the manual. Pages 4-63 have good pics so you can Identify the parts and how to assemble them correctly. Please use a torque wrench and torque it correctly so you don't have further issues.
We have seen this part on several 350 Honda ES go bad. Check to see if it's broken.
24620-HN5-M00
ARM, GEARSHIFT
24620-HN5-M00
ARM, GEARSHIFT
You will have to look it up. I can't get a pic to attach.