What's the ambient temperature? If the starter bendix has
petroleum lube on the spiral splines, it will stick at low temps and
won't engage. Lycoming specifies only silicone lube (which is dry after
application) on those splines. It's the screwing action of the spiral
that engages the gear. It doesn't take much stickiness to stop it
moving.
You shouldn't be able to disengage the starter by pulling though
by hand. It takes considerable RPM to do that.

Dan