My experience with these is that you have to have a good battery and it has to be fully charge. You should charge the battery for 24 hours before trying to start it. It the battery is good the starter should work. Your battery can have 12 plus volts and still not have the amps to make the starter engage and turn the engine. Hope this helps.
The forum is not Feaces Book so people are not compelled to answer posts in 3.5 nanno seconds.
Some pictures would have been good but at least you did post the end results.
Electric start mowers are not all that common down here, USA could be different.
Most of the regular posters do not post what they do not know and wil let a post like yours sit for a day or so then if there has been no answer, look up your mower ( if you have supplied enough information to do so ) and make a diagnosis.
I rather like dry Tri-Flow and it is good to know you can get the product over there under the same name as I have recommended it a few times.IT is particularly good with the multi stranded throttle cables Honda has a habit of using and the wheels on the self propelled.
Your description is typical of all these mowers and yes it is always the coil mount stud that pulls out.
A single drop of Blue or Green locktite will fix that problem and a dab of anti sieze on the other end of the stud work wonders.
You will probably find a lip on the spiral on then Bendix which is stopping the full travel of the pinion and a bit of Dremmel work to polish the lip off works wonders as well.
I fit Hawker Cyclon spiral cell batteries to electric start push mowers as they tend not to self discharge and will hold full voltage under quite heavy loads.