...how does changing the pickup height change the string height issue?
My mistake, I misunderstood your question and thought you had a clearance problem over the pickup.
Gibson specs for a Les Paul is 6/64ths
That sounds a little bit too high for an electric guitar. (and it's 64th because such rulers are in 64th).
Typically an acoustic guitar is setup to 6/64th at the low E and 4/64th at the high E-string at the 12th fret.
A good place to start for the action of a Les Paul type guitar is at the 12th fret (with no capo on) 5/64th" or 4/64″ on the low E string and 4/64th or 3/64″ on the high E.
Or for our European friends on the low E string a max of 2mm and on the high E String a clearance of max 1.5mm.
Mind that all depends on the players preferences. A hard strummer or a bluegrass picker might like the action relatively high, in order to avoid string-buzzing. Many jazz guitarists prefer low action, which is friendlier to
fingers playing fast scale runs and numerous, rapid, frequently complex chord changes. Lower action is more likely to produce string-buzzing.
So this would be o.k. for many, or a little bit too high for some. Depends also what string gauge you use and how hard you bend strings:
Before you adjust the bridge height you should however see if the neck is straight and correct if necessary with the truss rod nut.
Stick a capo on the first fret, press down on the 15th fret and then use the straight string as a guide to see how bent forward the neck is. The feeler gauge is put over the eighth or ninth fret underneath the string - .007” is the magic clearance, which is like a thin business card or two sheets of paper. Any smaller than that clearance and the string will bang onto the frets when played; much larger than that and the string’s height will be excessive.
With your action set and test played for buzzing, it’s time to set the intonation.
And at the very end set the pickup height:
https://www.premierguitar.com/articles/22114-mod-garage-humbucker-height-adjustment
Ralf