just a suggestion. check your battery again. there is a non zero possibility that the key START switch is borked or even the starter solenoid, and thus full energy is being applied to flywheel starter, thus it remains extended. This would be very rare that this happens, but I would still check to make sure those two devices are not borked and caused the problem to begin with. Of the few times I've seen or read a flywheel starter solenoid to pop out and remain out, is usually related to some other root cause. But then it could just be a flywheel starter that slammed against the flywheel and hit a tooth instead of a groove and has caused the gear bearing or shaft to bend hard...which isn't common,
as others have commented, the cheap flywheel starters are not cheap but are better quality. If cost is a concern, it would a good idea to run through the entire system before installing another one that might fail due to an unknown root cause.
note: I have had pretty good results with cheap flywheel starters. I am not going to say they are as good as the OEM and third party equivalents, but in a pinch and tight budget (or a customer on a budget and knows I cannot warranty the part replacement), I've gone that route. For my own personal commercial equipment ...always top quality Flywheel starters..And that can be serviced...such as the motor...or the brushes as others have commented.
just throwing out some ideas to help.