(This answer would not have been possible without the answers of Nathan Knutson and Argenti Apparatus.)
(I'm still not certain that this is the correct issue, but I had been wrestling with this adjustment for some time, and the issue has now evaporated once I used the solution below.)
An appropriate setting of the H limit screw is impossible if the barrel adjuster is malfunctioning.
It is not malfunctioning with a capital M, just a minor malfunction.
It's hard to see that that's the issue if one is just looking right/left/above/below the front derailleur.
If you have trouble setting the H limit screw, make sure that you are actually able to add tension in the FD cable through the barrel adjustment.
This answer shows the parts inside the barrel. It's an elementary device. But you may have the illusion that you are adding tension if the cable housing slips/glides on the metal part. The cause could be arbitrary (a tiny bit of oil, ..). Importantly, these two parts are designed to slip on each other. Again, as the answer there mentions, if the two parts do not slip against each other, the user could apply extra force and either keep adding undue torsion on the cable housing, or else simply strip the thread.
Disassemble, wipe any possible dirt/oil. Now make sure (visually) that the metal part does indeed start to protrude when you expect it to.
Now I'm very curious, is deception from a barrel adjuster a common problem? We're no longer in Q&A mode but in fireside chat mode. Add a comment or suggest an alternative medium.
Why? / Illustration:
Perhaps a diagram illustrates the issue better.
The center line is the cable. We're no longer talking specifically about the FD cable, only about cables with barrel adjusters in their middle, not an edge.
A and D are the cable housing. Notice that at all times the cable is under tension, but the cable housing is under compression. For derailleurs, the amount of tension/compression depends on the cog or chainring chosen. The bigger the cog/chainring, the bigger the tension.
C is the barrel adjuster. That's the part you grip. It has female threads. B is the part with the male threads. When B and C are fully threaded, the user cannot even see part B. B disappears inside C.
Here is the important part: this assembly works by ensuring that there is less friction on the threads (between B and C) than between A and B as well as between C and D.
When you turn the barrel adjuster, the relatively high friction between A/D (plastic) and B/C (metal) is what allows you to add or remove tension.
If any grit made its way into the B-C joint, or if any grease made its way to the A-B or to the C-D joint, the entire set-up will no longer work.
It is always prudent to clean the A-B and the C-D joint. No oil should be there. You want as much friction as plastic-against-metal will allow. (If you're following and want to make sure you never strip the threads, you do not want to add a drop of glue in there! The design includes slippage.)
I'm not so sure whether it's prudent, if you suspect that the B-C joint is somewhat rough, to add a drop of oil or a hint of grease. Adding grease there may invite an unwanted turn, either accidentally or from the repeated vibration when riding.