Simply try pulling the cable tight with your hand and see if it goes into 1st when pedaling. IF it doesn't, it's not the shifter.
Cable tension doesn't really increase that much until you try forcing it against the limit screw.

It sounds like you have the L screw on the RDER screwed in too much preventing the RDER from moving that last little bit.
You might try screwing it out 1/4 turn at at time and testing. Keep track of how much so you can reset it to where it was in case that doesn't fix it.
I have a hunch it will.