"To hell with logic, boosting stab instantly is BORING, but boosting it incrementally in a time-limited fashion without the introduction of any additional strategic consideration would be EXCITING".
Lol, no.
How's that illogical to be unable to turn in 1 day from an unstable state where people are ready to rebel, question the ruler's legitimacy,... to a very stable state where everyone is happy with the government and working hard? The current design is illogical, otherwise nations would have never fall because monarchs would just press buttons.
The question "what purpose does stability serve" in this game is a valid one. It is an ADM sink, offset by the state of the empire, significant only insofar as its impact on rebellions and the ability to declare wars/refuse deals. Time limiting it does nothing for the strategy of the game except increase the chance of rebellion should you get unlucky. Its present design is to provide incentive to manage unity and overextension. OP's suggestion adds nothing significant to the decision making process.
Although this is a personnal point of view, you make it sound as a truth. When I westernize, I expect to get rebelions all over the place (as is implied by events about people resisting westernization,...), but with the current design, I just don't. I press the magical stab button, and everyone forgets about my innovative policy. It should be a slow process, with a tough begining, and things smoothening over time, which is what a limit to stab raising would provide. Same for other situations; a stab hit during war and going in the negative could be an incentive to peace out quickly rather than suing more favorable terms.
A similar system for WE would make for shorter wars, rather than the old total war system; if I were limited in how much I could lower it, I would have to sue for peace at some point, or face the risk of dangerous rebelions. You would have to weight the possible gains and the possible losses; that would be a more logical soft cap to expansion; if you could manage your own business fine, you could keep expanding, but if you were to be a bit too ambitious and involved in a tough war, then you would have to pay the price for it. The current design just doesn't make sense; there are many mechanics that on their own could mimic disaster system, yet rather than having a "disaster-like" scenario happening with low stab while involved in a messy war with low legitimacy (rebelions because of high unrest; no need for dedicated events), it makes those modifiers insignificant and implements revolts through events.
By the way, the "time-limited fashion" used to be how it worked in EUIII, with similar modifiers (revolt risk,...); you had to choose how much gold to spend monthly for stab and tech (rather than MP), so it's not really a suggestion "without additional strategical consideration", as it would be a direct mimic of some of EUIII mechanics (one of which were better in my view). I'm guessing it has been changed to make things easier rather than just a "MP sink", but it failed at it and both stability and war exhaustion aren't modifiers to be feared anymore. Even inflation; I can be the Aztecs, get like +0.15 monthly inflation and still keep sitting bellow 2. Might as well outright remove inflation and just pop an event "you're mining gold! pay 50 ADM to proceed!" from time to time.
What I mean is that the game tries to be complex, has many modifiers, but in the end it just add rules on top to make this apparent complexity insignificant. I could go on about how I either sit on 0 prestige at start or 100 prestige 50 years in, but this is getting off topic.