This is the bottom of the mirror box for the Pentax MX. The red arrow is pointing to a post that interacts with the winding mechanism. As you wind the camera, a long arm pushes this post to the left (front of camera). You should be able to see this happen with the bottom plate of the camera removed. If the camera is operating correctly, only the left end of the spring (green arrow) should move. The right end (blue arrow) should stay in place, loading the spring so it can quickly flip up the mirror. On your camera both are moving, so the mirror moves up as you wind the camera. To see why, we need to look at the side of the mirror box.
The red and blue arrows are pointing to the same things, just seen from a different point of view. The purple arrow is pointing at the little lever that sticks out into the lens mount. That's what you wiggled to get it working but normally it's what stops down the aperture in the lens
. The pink arrow is pointing to the lever that flips up the mirror. The yellow arrow is pointing to a latch that is supposed to hold the mirror up mechanism in place as you wind the camera. I think this is where the problem is. There is a small bias or return spring which is supposed to return the latch to its proper position after the shutter is pressed (pressing the shutter moves the latch allowing the mirror to flip up before firing the shutter). The latch on your camera is probably too gunked up and the spring is not strong enough to return it to the home position. Messing with the stop down lever (purple arrow) probably provides just enough extra force to get it to return, at least for one cycle.
I hesitate to recommend this, but you might be able to free up that latch pivot by carefully applying some solvent (Isopropyl alcohol) through the gap around the stop down lever. Then run it through a bunch of cycles to work it in. But even if you get it firing consistently, it's likely just a temporary solution. The proper fix is a disassembly of the camera so you can clean away all the old gunk. Look up Erik Hendrickson at pentaxs.com. His price for a standard CLA are pretty reasonable.