As I'm sure you are aware, sleep deprivation is NOT primarily about the physical detriment but more affects your mental and cognitive functions. The exhaustion table is too heavily focused on the physical aspect to really apply. As all long-term sleep deprivation studies have shown cognitive impairment is high and, while you might want to simply apply Exhaustion, I don't think speed halved is really appropriate, or hit point max halved, are directly relatable.
The other issue is Sleep deprivation is more heavily a "willpower" focus, which now falls under the purview of wisdom saving throws. I can understand why you wish you could simplify it down to a Con check, but it's really not a physical endeavour to stay awake, but a mental one. Feeling "so tired" because of a lack of sleep is more because your brain is struggling to cope, as opposed to your body.
If you want a simple thing, I'd go with:
You can stay awake for 24 hours with no effect, after which a DC 15 Wisdom check is required to stay awake. After 48 hours awake you suffer disadvantage on all checks and cannot concentrate on spells.