My snapper was a one pull. The bigger motor made a big difference. I have to run an extension cord to mine but don't mind since it allows it to fire right away.I was having problems starting my single stage Toro for a couple of years. The first start of the year would take about half an hour. After that it was about a 5 minute process. I felt so dumb when I noticed that the primer bulb was split all along the bottom so it was doing nothing. About $2-3 for the part and about 5 minutes of cursing getting the thing on and I was back up and running. Starts like a champ now.
My electric start worked for about 5-7 years and then is started sounding like a grinder wheel so I just stopped using it. The pull start is fine for me. I never checked into what it would take to get the starter working properly again. It is of limited usefulness anyway since it is powered by plugging a short cord into a wall outlet
What you described is the starter being worn or it moving away from the fly wheel a bit. What mine does now. I need to replace this one (think its the second one) since it doesn't seem to be working right. Last time it was about 90 bucks for a new starter, guessing it has doubled since then.
EDIT: Just checked online and for 70 I can get a new one. Parts getting cheaper is not common.