How is it artificial ? People don't get tired walking hundreds of miles workout sleep ? I guess health is artificial and unnecessary too.
Then under you system, you shouldn't be able to "slow travel" without taking the same sleep breaks.
Your mistake is in thinking of fast travel as "fast". Not your fault, it wasn't named properly. It should have been called "Automatic Travel" instead of "Fast Travel". It's not any faster than running there (in game time), and generally, except for time spent fighting (which you don't really have to do anyway, I've yet to run into something along the road I couldn't get away from with a horse), is actually slower. In other words, in most cases, more game time will have elapsed by "fast traveling" than by running, especially if you use a horse. From the character's perspective, it's really "slow travel". It's only "fast" travel from the perspective of the player, as it happens within a few earth time seconds (or more, depending on how fast your load times are).
Your character doesn't gain anything by you using fast travel, so why should he need more rest than if you were using the controls to make him run?