2 things....
First is after play, I wipe the chalk off with the palm of my hand. And I vigorously rub the tip with the palm until I feel a heat build up. I'm using the oil in my hand to condition the tip after every use. When I get the shaft back out, the tip holds chalk like it was magnetic.
Does anyone else do this to their tips? I've heard of people using saliva to condition the leather.
From a materials standpoint, these two are not advisable. The objective of burnishing the sides is to flatten the grain by forming a barrier through the combination of moisture and friction. You can get a little bit of burnishing just through friction, which is what is happening when you rub the playing area of the tip; has the opposite effect of scuffing a tip and reduces surface area for the tip to hold onto chalk.
I'm hesitant to condition leather tips. Cue tips are made of vegetable tanned leather, and reacts to every form of moisture applied. On the far end, you can reshape vegetable tanned leather by submerging it in water. Once it dries out, it hardens and retains the shape it was in, most easily imaginable in leather armor. In this case, neatsfoot oil may be used to make the leather supple, but too supple isn't a good thing for play, though good for longevity. It would be even less advisable to do so with layered tips where the interstitial adhesive may become compromised.
This is where milk duds come in - you start out with a veg tan tip that's pretty loose in grain (= more surface area to hold onto chalk), soak it so it's pliable, press it to reduce future mushrooming, and has the net effect of being hardened. The milk or whatever recipe should be optimized to condition the leather to prevent it from feeling too dry, but I've never come across any oils added because that's seemingly not the form of desired conditioning.
As far as Fats keeping them in his pocket, tanned isn't quite the right word, like how film doesn't develop any further after it's stable enough to be in the light. Through contact to dirt and oils, it develops a patina, which puts it in a stable, pliable state where it holds 'memory' of how it's moved around; a well made wallet or boot being a familiar example. Hand oils will color, but not interact with the leather for the purpose of longevity to a significant degree.
Here's a quick test comparing chalk adhesion with burnished/unburnished leather:
The darker side is burnished and oiled
Only the burnished portion where the grain is coming back up is catching chalk.
Before Chalking, shiny side is burnished like the sides of the tip
After
