"""""contrary to Moly which is mostly shed in the first few cm as the bearing surface of the bullet gets rid of it due to friction """"""""""""""""
First, I respectfully don't agree with you above statement.
Should a bullet be correctly molycoated, one can recover shot bullets which are still coated inside the land prints and everywhere else.
I do pay a lot of attention to degreasing THEN removing copper oxyde on bullets prior to molycoat in a rotating tumbler. Even if bullets looks shining new. I stop tumbling when tiny MoS2 concretions / scales start building on the jacket. These little concretions are easily wiped off a coton or nylon cloth.
Now with regards to barrel conditionning, I have been using lab fine grade MoS2 on patches after cleaning/drying barrels in .30 HBR and 308 Win with good success.
I use it the dry way, preparing powder saturated patches at home. Then after cleaning and drying/degreasing the barrel, I will make 10-15 strokes with the moly'ed patch, then run a dry clean patch that will not fit too hard in the barrel to wipe off the excess and go shooting.
Please note I do not use this technique in 6mm barrels. I tried, was not satisfied, I consider them too small to reach a good spreading of the moly powder, most particularly in the groove/land "corner" area. Maybe some saturated vfg cylindrical felt pads may be used in small barrels. I dunnow.
I also tried some liquid carriers as acetone or gasoline for the moly powder. Works a tad better in small diameter barrels. Issue is then to clean the excess you get in groove/land "corner" because of gravity. Never let the carrier fully evaporate.
I am also using a gear box oil in my cleaning compound. That transmission oil includes some calcium sulfonate with is a very good high pressure lubricant. Calcium sulfonate particles will, as moly, fill the steel grain interstices (not the bath tubb size pits you get close to the chamber).
After "seasoning" a barrel that way, first bullet slide pressure + 4000 bars gas pressure will push moly or calcium sulfo particles in steel grains interstice and it will stay here almost forever.
With regard to "alligator skin" at the throat and forward, that is, in my mind, mostly due to surface nitridation of barrel Fe atoms + thermal stress, leading to cracks and pits. Should you look carefully in there, you might see that your alligator most probably lost a scale here and there. That will never forever get filled with moly or whatever. That will increase bullet scrapping, aka copper fouling.
It's maybe time to go in there with flitz or any kind of barrel steel abrasive compound to "round the edges" in order to limit bullet scrapping. .... And get a new barrel for the match.