It seems that the Devs never really thought through the whole Exhaustion mechanic much. While it appears in the rules system, it is under utilized, there just aren't many places where it comes up. It also is somewhat baffling that it is a harsher condition to recover from in some ways than death (3rd level Revivify versus the 5th level Greater Restoration). I once considered having some of the underutilized necromancy spells apply the condition on a failed save, but the wonky ordering of what is penalized as the condition progresses seemed to really only matter to PCs most of the time.
It would be better, imho, if the condition progression for Exhaustion were modified in some way to make the earlier levels more even while having Lesser Restoration remove a level. This would allow the mechanic to be used more often and in more ways.
As for the Exhaustion for a death save mechanic, I suppose it depends on the play style and table mostly. Some will find it enhances the game, others will see it lead to the dreaded 5MW and such.