All gears shift smoothly, but won’t shift into 1st gear/large cog. Help?
From what you describe, it seems that pushing the derailleur inwards manually to get it onto the largest sprocket was tightening the shifter cable - that is suspicious, it should have loosened the cable.

First thing to do is disconnect the shift cable from the derailleur and check three things:


The derailleur will swing through its full range
The shifter will click through all gears.
The cable travels freely. You should be able to get hold of the shifter and derailleur ends of the cable and pull it back and forth.


If anything is hanging up or jamming, that is where you need to look for problems. I'd also check the shifter cable routing and make sure there are no cable and housing are where they should be and there are no kinks.