from the general read, I suspect that the flywheel starter solenoid is either GOING BAD OR, it's just got crude withing it.
here is what I would do...spray some wd-40 into the neck of the flywheel starter solenoid...saturate it. Let it sit for a while.
now since you have been using your mower battery to try and start it, you could be draining the state of charge quite a bit...and that fake you out thinking the battery is okay, but forgetting that every single time you try to start and it does not run and being a charging cycle, then you are left with less and less energy...So put a battery charger on it...just to be sure. let that sit for an hour..let the wd-40 does its magic. After an hour, take the charger off, and CAREFULLY WITH THE PLASTIC HANDLE OF A SCREW DRIVER, give that starter solenoid a few solid TAPS...NOT STRIKES...just some light taps. Tap it on the body....tap it at the gear...side to side and up and down.
now try to start it.
if the symptoms go away, more than likely the cause is a flywheel starter that has accumulated some rust of grass varnish...internally you just washed that crude out with the wd-40 and the taps loosened some junk up.
here is what I would not do: repeatedly try to start the mower in this condition. if the flywheel starter is not the problem, then you can definitely overheat and overwork a flywheel starter solenoid to the point of failure by trying to rotate the engine over and over and over.
as other have pointed out...there could be another root cause of this problem. One easy way to make sure the engine isn't the issue is to take spark plugs out and rotate flywheel by HAND...if it rotates easily with little resistance, you can rule out the engine as the problem. if sparks plugs out and it's hard to rotate by hand, then you need to start looking at the engine...