John,
The starter consists of 2 parts.
1st; the motor assembly....this SPINS the starter gear only.
2nd; the solenoid......this ENGAGES the starter GEAR to contact the flywheel teeth.

So, if the gear is indeed spinning, this WILL indeed HEAT up the starter motor. Remember, do not run the starter excessively. This WILL overheat and burn it out, ie; the windings in the starter motor itself.

The difficulty you seem to be experiencing appears to be the engagement of the gear and the flywheel. Thus, a broken tooth on the flywheel could cause your trouble. an example is that when the engine shuts down, the inherent balance and compression etc. cause the engine to stop moving in a position which occurs somewhat regularly. ie; the flywheel stopping in a couple of common stopping points.....one of which MAY be on a chipped or broken tooth.

However, the MOST COMMON reason for the starter gear to NOT engage is a malfunction of the solenoid. All that a solenoid is is an electrically actuated PLUNGER that pushes the gear forward and retracts when the electricity is shut off. The plunger may STICK in its barrel, and or the pivot point at the gear may be stuck.

Anyway, enough of the theory. Save receipts. (i know that you have)....if you do not get the gears to mesh, (the hammer while turning the key is an EXCELLENT idea) then remove (sorry) the unit for a bench test....This should allow you to actually see the plunger moving in and out.....Even with a BOSCH reman. you may just have a defective unit.....Beware of the so called warrenty with ELECTRICAL components.....perhaps you know of a shop which would say that they installed it PROFESSIONALLY.

How is the POLICE CAR? Fast ......BEWARE of the ford POS transmission. The other caveat is the notorious heater box (now this way in MOST newer autos )

Watch out for speeding tix.

Best wishes always, spo.