I think you can't engage the clutch by pushing the cable, it's the spring in the clutch that pushes the plates together. If the spring does not work as it should, there would be no way to compress the plates one against another.
The cable is probably causing friction somewhere, and it does not allow the spring to pull it back and engage the plates fully. Lubing should help. Check for fraying or any damage - if cable is causing that much friction it might be damaged, that spring is pretty strong.
Here is the link to Kelly's DIY about cable lubing.
And
here is a link to a very thorough clutch plates replacement DIY. Inside you will find some more useful links to clutch lever / cable issues some of us have run into.