Possible causes:
FD adjustment (limit screws)
bent derailleur cage, or mount
cable friction
failed cable or housing
Sounds like you're checking the limit screws already. Do I understand correctly that the FD can reach the big ring if the H limit screw set properly, but for some reason the force required to shift it there is very high?
Shifters can wear out (usually worn plastic internals), and Tourney shifters are less than $20 each.
Just trying to diagnose that the shifter is indeed the problem. I.e., a new shifter won't solve the problem if it's caused by improper set up/tuning.
Park Tool has a good guide on their website:
https://www.parktool.com/blog/repair...eur-adjustment. Best of luck.