I would imagine there must be some sort of way to test the pin connectors on the card with a voltmeter of sorts, however I do not know the specifics. But if we think about this logically, the problem should be with the fan itself, rather than the heatsink or the GPU card. When you press the connection to make it tighter, you are forcing the stripped end of the wiring inside the connector to make a more solid contact and thus allowing the flow of electricity to continue. But as the male connectors (pins on the gpu) should be conductive throughout the surface area, the only way to make sense of your results would be that the soldered connection is decaying and thus compacting the pins back into their respective slots. Seeing as this hardly make sense even as I type it, I would say the fault lies in the fan. Additionally, it appears partial fault could be resulting from bad bearings within the fan. Seems the recommendation is to remove the sticker in the center and add a drop of oil or two. However, at this point that may be redundant as the fan may be dead.
Sorry that I cannot think of any more cost efficient method to narrow the solution down one way or another. Best of luck with coming to a resolution. - Rube