I can pull off a pinch harmonic on the thinner strings (1-3, usually 4th) but the 5th and 6th got me picking just the fretted note.

Is there something I don't know? Maybe I should use more thumb, hit the string with thumb later or quicker?

idk... i do them all the same way... it took me a while to get those too but id say just practice alot and you should get them
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it works the other way with me, can do it all fine on the 4 low strings, only just on the 2nd, cant on the 1st, i think maybe for the lower ones you dig in a bit more? but i cant really say
it really depends where u hit them between the pickups, just hit different spots and memorise where each sound is
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yeah i definately use more thumb on the thicker strings, i dont know if that's just a style thing or if u actually need 2 do it tho......... what i do is just practice my pinchies by doing ascending and descending chromatic runs all down strokes and try to make sure i get a pinch outa each note, worked 4 me now i can hit them pretty much anywhere........ i find that they dont sound as cool on the thin strings once i get above the 14th fret tho.......
I have no trouble on any of the strings, it works fine but how it comes out depends on where you pick. I've been doing that kind of harmonics for nearly 30 years, so it's second nature now, it's a lot more difficult first learning. I find for dual humbucker setups like Les Pauls I get the best squealies between the two pickups, about 1/2" off the bridge side of the pickup. That's what it sounds like Billy Gibbons uses most.

Another good spot is just at the edge of the fretboard, you get a more mellow sounding harmonic there. Once you get the hang of it, try moving down the neck to directly above the last or next to last fret, those are hard to get because you are really close to the fretboard, but when they work it's wicked.

Technique...I hold the pick with just a hint of pick sticking out of my fingers and lay the thumb flat as possible so it catches the string well and the index finger curled around the pick. If I'm warmed up good, doing well and having a good night I can actually get that type of harmonic on an acoustic without a pick, using just my thumb and no fingernail, by laying my thumb flat against the string and picking it, letting the knuckle create the harmonic, but it's very difficult and never consistent. Found that one out by surprise, held my thumb just right one day and there it was...I've been practicing it, but in 5 or 6 years of working on it there's no such thing as consistency yet.
Hmmm...I wonder what this button does...