I've a Honda 3011 that refuses to play nicely. It does not like to start.
It would not start and would just respond with the click.
If I turn the flywheel or the starter gear, it is more likely to start (though still less than 50%). It lifts the starter gear and engages properly once or twice, but then go back to not engaging.
This is obviously an impractical way of starting the mower to do the lawn.
Would a bad solenoid cause this sort of behavior, or is that more of an all or nothing proposition?
If the solenoid is mounted on the starter ( like a car starter ) then yes the solenoid .
If the solenoid is remote then it will be the bendix gears on the top of the starter which are a wear ( thus replace ) item.
Don't bother they will not interchange.
The B & S will be imperial and the Honda will be metric.
A CAR electrician will be able to repair your starter ( well get the parts to repair it at least ).
Mower shops will only sell you an entire starter $$$$$$$