I don't know the answer to what is wrong for sure, but there is a logical answer to this question, which may help guide where to look. The shutter release on cameras with a half-press is often actually two switches (electrical or mechanical). In the case of the F3 the first switch on half-press powers the camera on, and the second switch fires the shutter. In your camera the first switch in the body isn't working, so the camera doesn't power on. But if you power it on by half-pressing the MD-4, then the second switch in the body can close its contact and fire the shutter.
It might seem like pressing the body release all the way should provide power to the camera and allow it to fire, but for whatever reason the F3 isn't wired that way.
Here is a screenshot from the repair manual showing mechanically how the two switches work. (Ignore the "troubleshooting" paragraph as that's related to a different item on the same page.) It is possible that the problem is not in the mechanics, but somewhere in the electrical contacts: for example if part 11 is disconnected from the power circuit, but part 12 is still connected to the release electromagnet, then it's possible that pressing the button all the way would not power up the camera, but would still fire the camera if powered up from the MD-4.