Difficult to understand what parts you are referring to since those are not the proper terms. Nothing gets pushed into place to engage with the central chronograph wheel for the minute counter. There is a finger on the chronograph wheel that flips the minute counter driving wheel, and that wheel in turn flips the minute counter. The minute counter driving wheel is in a fixed position on the plate if that is what you are referring to as the "intermediate counter" so it does not move in to engage with the finger on the chronograph wheel.
Typically if the minute counter is not flipping over the finger on the chronograph wheel has somehow gone out of adjustment (it's a piece of bent steel). It could also be that the minute counter jumper is out of position and jamming the minute counter, but if so I would expect the watch to stop when the minute counter tries to advance.
Here is a video showing the minute counter failing to trip on a 7750 based watch. The finger was bent out of position so although it starts to move the minute counter drive wheel, it doesn't fully flip it over since only the very last bit of the finger contacts the wheel:
Here it is after adjustment - ideally the finger should contact the tooth of the minute counter driving wheel at the midpoint of the flat on the finger as shown. The finger then drives the wheel far enough to flip the counter, and then contacts the subsequent tooth of the drive wheel and pushes it just far enough to take up the backlash between the teeth of the drive wheel and the minute counter - you can see that taken up near the bottom of the screen. So this is perfectly adjusted:
The watch really needs to be looked at in order to fully diagnose the issue, but the finger being bent out of shape would be the most logical scenario given the OP's description of the problem.
Cheers, Al
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