"Turning over slowly" and "Not engaging" are two very different issues. Which is it? The reason it was suggest to spin without engage was to test the starter motor. This however does not test the starter motor OR it's connections under load. Under load, operation can change.
The engagement is done by your foot and the linkage. Not only can you test the motor by using the switch only, (as Jerry said) you can test the engagement by disconnecting the battery and operating the engagement, inspect the pinion gear on starter, turn it to let it engage the flywheel to see what you can see.
