I propose that one ought to track fatigue, that this is best represented as a saving-throw like mechanism as opposed to ablative hit points or through making CON checks, and that the fatigue check modifier or target number ought to vary by class and level.
Fatigue is worth tracking because it forms a way to naturally pace the rate at which PCs accomplish things, especially overland travel, dungeon exploration, and combat. As fatigue mounts, the players are encouraged to stop exploring temporarily until they can rest up. PCs can track this situation as it develops, rather than simply waiting for a declaration from the referee that they are too tired to continue or waiting for the wizard's spells to exhaust themselves or until their hit points are nearly depleted.
Tracking fatigue also allows one to have it interact with other mechanics. For example, I've been thinking for a while of having spellcasters make fatigue checks after casting spells rather than having a certain number of available slots to expend. This is the other side of Vancian magic, one that is not well-represented in the rules - the physical and mental demand of memorising and using the spells. A wizard who has cast all their spells is not otherwise impaired in D&D, though I've often seen the reasoning in-game be that the wizard is tired. Having a fatigue check would translate this impressionistic declaration into a concrete status change.
The idea is that a wizard would memorise a small number of spells from each level of spell they know, and be able to cast any of them as many times as they pleased, with the condition that they have to make a pass a fatigue check before each one. Failing a fatigue check could have several possible outcomes based on the referee's taste. The first is that failing a fatigue check could cause the spell to misfire. The second is that each failed fatigue check closes off a level of spells, starting from the highest known to the wizard and progressing down until the wizard is too exhausted to cast even the weakest 1st level spell. I'm sure most of you can come up with other ideas as you prefer.
Other abilities could be linked to the fatigue check. Specifically, I would link clerical turning to it, along with druid shapechanging and lay on hands. Rather than a finite number of uses that must be tracked for these abilities, fatigue checks would provide a way of passing them. Fighters in particular wouldn't have any abilities that required them to make fatigue checks to use, reinforcing the game concept that fighters are characters whose abilities do not "run out" despite not being as fantastical as a wizard's spells or druid's shapechanging.
I prefer the concept of a saving throw-like mechanic over points and CON checks. Fatigue points are extremely fiddly, and integrating them in the above way causes the balance of the mechanics to go wonky. Casters at high levels need tons of fatigue points to expend to cast their spells, so many that fatigue becomes trivial for mundane characters like fighters and thieves who are not otherwise expending their points. It also requires yet another resource to be tracked, one that will probably change even more than HP do.
CON checks are one possibility, but I favour fatigue check target numbers varying by class and level. CON varies more by character than by class, and is hard to improve or change. It means that the quality of the wizard is based more on their stats than their class or level, something I would prefer to avoid.
The value of varying by class and level has been touched on above. It gives yet another point of differentiation between fighters, thieves, and mundane classes from the supernatural classes, one that favours the mundane classes (who have better scores and make fewer checks). Higher level PCs will tire less rapidly than low level ones, having become accustomed to hard living, just as their saving throws improve to represent a greater instinct for avoiding certain doom.
I favour a system whereby there are two kinds of fatigue checks a PC can make. Let's call them minor and major. "Minor" checks are ones whereby a PC is doing something strenuous and needs to stop to catch their breath or recover from a sudden failure of endurance or will. A PC who rolls under their fatigue number must stop what they are doing (fleeing from monsters, casting a spell climbing the rope up the cliff, being engaged in a battle of will with a malign psychic parasite) and rest for at least 1 minute. That's the only penalty in most cases, except for spellcasting, where the caster also loses access to their highest level of spells (fatigue checks for any reason cause this). Minor checks are meant to be used in situations where constant, strenuous effort is required, and ceasing it will have negative consequences (being caught by the monsters, needing to find a place to rest while suspended in mid air having the psychic parasite seize control of your mind).
"Major" checks represent physical and mental exertions which could potentially require several hours of rest to recover from. Marching overland, translating ancient writings, training, building things, and exploring a dungeon carefully are all examples. The PC must make a fatigue check after doing these things. Failure means they are exhausted. Until they can rest for 4 hours, they automatically fail all other fatigue checks (major and minor). The idea is that major checks are for extended activities where the PCs are undergoing constant exertion, but where failure isn't inherently problematic, and they can rest at least temporarily before going on. The fatigue check represents their state at the end of the activity.
Fatigue checks can also be used in other circumstances. For example, I would use them to determine whether or not a character wakes up when the camp is attacked - get too tired and you'll sleep like the dead.