Left shifter is not catching/gets stuck when there is cable tension

I need help with my left shifter. When I have the cable attached to the derailleur and have even the slightest tension applied, the shifter won’t shift to the largest cog (I have double front chainrings). It feels like the shifter is stuck and I can’t push the lever in. But when the cable is detached and there is no tension whatsoever, the shifting works perfectly fine.

Some background to how this happened:

I was noticing some chain rub when I was on my lowest gears (small cog front and largest cog back) and I figured it was something that I can fix easily. I followed the GCN and Parktool video and detached the cable from the derailleur clamp. Then I adjusted the lower limit screw. I then reattached the cable, and from then I noticed that I couldn’t shift to the larger cog. Both the GCN and Parktool video said to adjust the tension, and that’s what I tried. From then, I spiralled into a rabbit hole. Worth saying that I had no problem shifting between the cogs before this.

What I’ve done so far:

• Started all the way from scratch – derailleur and cage height is good
• The derailleur is working properly, I can move it manually without
  problem
• Loosed the H-limit screw
• Adjusted the cable tension – from slack to very, very taut
• Pulled back the lever hoods- I saw that without tension, the shifting mechanism worked perfectly fine. It was going all the way around and pulling the cable properly. When tension was applied, it was getting "stuck", and can’t push the lever all the way
• Oiled the cable and checked for fraying. It looked fine.
• Flushed the shifter with degreaser.

So now, I’m at the end of my rope. I just can’t believe the shifter broke for no reason. Is there something I’m missing?