I would not worry about any engine damage if you only had a couple lean runs. A couple lean runs does not help your motor but certainly will not ruin it.
Even a new engine with a couple runaways on it will run fine but may not have the life span as an engine that has not had runaways.

Im not saying to go and run your engine lean and expect to have long service from it. Im just saying that some opinions exaggerate what is actaully happening with these little engines.

In my experience I have run several engines very lean on purpose while racing and still got 4 + gallons out of em. I have never worried about how the engine cools either and I seem to get plenty of service from all my car and plane engines.

There may be other issues as to why your engine is not starting.

Dbow