Success gentlemen, I was able to remove the transmission from the engine! It was stubborn, requiring me to shake it loose from underneath the car, and my idea of utilizing an old tire to catch the transmission centered around the jack, really paid off.
As a side, I was able to take a look at the pressure plate and flywheel, while noting how small everything was compared to what I've recieved from Exedy. When I went to touch the flywheel, a ring gear fell to the ground.
From what I've seen from the shop manual, this setup is not stock. My guess is that this job has been done before. I did note however, that all of the teeth on the ring gear were still intact, making wonder why the engine would only engage when push started. The teeth on the starter looked fine too.
At any rate, I ran out of daylight and will start removing the pressure plate & clutch disc tomorrow. By then, I'm hoping to see some kind of evidence that indicates what I'd pondered earlier.