Who DARES disturb my slumber!
With a few exceptions like French and Soviet halftracks, only the US used rubber based tracks. The German halftracks and Pz II Ausf D used rubber padded but mainly steel tracks. The French, US and the rare Soviet halftracks used the Kégresse track system. That was basically a one piece rubber band reinforced by steel cables and spacer plates. The original concept was that a suspension unit and tracks could replace a truck's rear wheels. The downside was a lack of durability and if something tore the whole track had to be replaced. Due to their one piece construction, they were cheaper to manufacture (bigger and more expensive moulds but much, much less labour).
The other US rubber tracks were blocks of rubber with steel bars front and rear. The ends of the bars were fitted with steel connecting plates to join the rubber blocks. The bars passed side to side through the width of the tracks. In later designs you had 2 slightly separated rubber blocks (side by side) but sharing the same pair of bars. Like the more common all steel tracks, the rubber-steel shoes could be replaced individually. In service the original plain shoes lacked grip and chevrons were moulded as part of the rubber block or steel chevrons were inserted as part of the moulding process. All sorts of track pattern variations could be seen on M4 Shermans alone. The short production Grizzly, a Canadian M4A1 variant, replaced the tracks with their own all steel track (and drive sprocket). Some early M4A3E8 and M26 tanks had all steel plates, but were replaced by rubber-steel ones.
Pros? They were lighter, overall as easy to maintain/replace as all steel. There was a trade-off between steel usage and scarce rubber (especially after the Japanese seized Malaya). Overall they were easier on roads due to the extra cushioning effect of the rubber and quieter to boot. The early smooth or rubber chevron variants were extra easier on roads, though the later ones had chevrons or bars of steel for grip and had as many road tearing steel bits as you would see on a Orc's club in a fantasy film. They seemed overall to last for more miles, at least that's something the Soviets took note of. Due to their design they tended to curl up which helped when pulled up at the rear when moving forward, IE they had a little less "drag" than simple limp steel tracks.
Cons? The early tracks, allied to their narrowness, meant that they lacked grip. They were less durable in rocky or sharp gravel conditions. They burned. Their grip in ice was particularly bad, though steel tracks could become a cropper as well, as testified by the Germans adding mini-chevrons to their steel tracks.
It took time to fix the grip problem, but eventually the US must have been satisfied as rubber based or faced tracks continue to this day. Indeed many nations like the British who used steel tracks switched to rubber or rubber padded tracks post war. Nothing like the noise and having your roads torn up in peacetime to push you that way.
As to how you would reflect that in ASL, I really don't want to go there. I will note that despite the US's well deserved reputation for mechanically reliable vehicles, US halftracks have
Red MP. That I put down to the flimsy nature of the Kégresse suspension
and track, rather than simply due to the track alone.