So, on the real-life example, from back when I did long distance running, I'd agree that the first thing that hits is reduced speed. You go from running, to jogging, to walking, and just kind of keep moving until you've recovered enough to pick up the pace again.
Comparing on the idea of bailing, it takes a lot to get to the point where you can't keep moving at all, but I wouldn't be surprised if he wasn't as quick about the process by the end of the day as he was at the beginning.
At the same time, I don't see mental processes slowing down in the early stages, but definitely in the later stages. At the same time, I wouldn't characterize it as being more likely to fail (disadvantage), but rather taking longer for your brain to work. It may not matter if it takes longer for you to remember that historical fact, but it's much more likely to matter that it's taking you a lot longer to notice and react to the shadows moving in the trees up ahead.
Going on a tangent, I'd wonder about a version of Disadvantage where, if the lower value fails, but the higher value would have been a success, you have to take the consequences of failure, but then you still get the success. For the history check, it just means you're a bit groggy and slow to answer. For the perception check, you might end up getting ambushed because you were slow to notice things. For the animal handling check, you might be slow to recognize the signs that the dog was going to bite you, and you end up getting bit before you get it under control.
On the other hand, if you fail to jump the pit, then it doesn't matter if you get the success after the failure, because failure means the success couldn't happen at all. And if you applied this to attack rolls, hitting with the attack overturns the consequences of missing the attack, so it's again something that can't happen, and thus doesn't. In other words, it makes sense without breaking anything even if you apply this rule across the board.
So if you just treated disadvantage in general this way, it would largely be fine.
If you used that form of disadvantage, then I'd say the basic form in the rulebook is perfectly fine. It's where disadvantage often means failure, and you often can't reroll a check another time, where that first level of exhaustion is incredibly frustrating to deal with. At the same time... yeah, speed drops before you really start getting loopy or groggy.
Level 1: Speed halved
Level 2: Disadvantage on ability checks
Level 3: Disadvantage on attack rolls and saving throws
Level 4: Hit point maximum halved
Level 5: Speed reduced to 0
Level 6: Death
The only other thing I would consider would be that you could recover from a level 1 exhaustion with a short rest instead of a long rest. You can get to the point of dragging your feet in a relatively short time period, but it also only takes an hour or so of resting to get back to where you're ready to keep going. By the time you're getting loopy (disadvantage on ability checks), though, yeah, you need a good night's rest to get back in shape.
That would also make the rules about exhaustion from combat more reasonable. When half the party can expect to pick up a level of exhaustion from even a relatively trivial fight, and that means that most of the rest of the day of regular activity is wasted, people are going to actively avoid pushing forward, and take short rests as much as possible. If they can expect to recover from that first level of exhaustion from a short rest, though, they're more likely to push on a bit longer, particularly if they just have to deal with the less frustrating speed reduction, while still expecting to be able to be productive during the rest of the day.
Having just gotten through a campaign involving magic items that induced exhaustion, the constant need for a night's rest (or several) just dragged the entire adventure down, slowing everything down.