I would take a step back before applying a new rule to fix a problem.
You're experiencing a natural outcome of the combination of several factors:
1. Dropping to zero hit points results in a level of exhaustion
2. Your campaigns feature adventures with time pressure
3. Exhaustion takes multiple long rests to remove
So, to me, the issue is that combats you're featuring in adventures which have time pressures are resulting in players dropping to zero hit points.
That means the real root of the problem is one of three things. The first two are easy:
1. Your combats are too hard in adventures which feature time pressures.
2. Your time pressures fail to account for the difficulty of your encounters.
These first two are two sides of the same coin: "Is my adventure design working correctly and fairly?"
You've got direct control over is encounter difficulty and the time pressure timeline. If your players are all running their characters tactically well and they're spending a reasonable number of resources on in-combat healing, but unpreventable on unavoidable damage spikes are taking out characters more quickly than the PCs can react to, then your encounters are too hard for the adventure timeline. If your PCs are facing combats they could never avoid and are still struggling to meet the timeline because they need downtime for healing, your timeline may simply be unrealistic or unfair. If the PCs are facing mandatory combat encounters regardless of their actions and, in spite of spending meaningful resources to heal in combat and taking short rests, they're risking failure because of recovery time in that they absolutely will need to have taken in order to complete their goal (i.e., the PCs can only beat the final boss if they're at 100%) then your adventure design may be fatally flawed.
The third potential root of the problem is more interesting:
3. Your players are unable or unwilling to be more conservative with healing in combat to avoid dropping to zero hit points in adventures which feature time pressures. In other words, they don't adjust their tactics to account for different situations.
If your players are truthfully just continuing to run their characters as recklessly as they do when there is no time pressure, then at some point they need to just accept the results. If you think the players could completely avoid their characters dropping to zero hit points if they changed tactics or strategy, then this is exactly what I would do. You can't control how liberal or conservative your players are with healing potions or healing spells, but if you honestly believe the party is plainly being too conservative with healing spells or plainly being too reckless with how they're approaching combat, then they should fail. You can try to teach them with unsubtle hints. For example, after dealing a heavy blow to a character, remind the party that the cost of exhaustion will slow them down significantly. You may also provide some bonus healing potions or scrolls at the beginning of a time pressured adventure "on loan until the quest is complete" to give them both a hint and a buffer.
If the players are risking skipping a short rest in order to try to save time and the characters suffer exhaustion because of it, that's their choice. Risking losing time because they have to double long rest or risking characters with significant penalties in combat is the balancing cost of that risk. If there's no teeth to the risk, there's no glory in enduring it, either. The players should know they're under time pressure (but you can remind them!). The players should know they'll be slowed down even more if they're too reckless (but you can remind them!). IMO, modifying exhaustion recovery just rewards reckless play, and the whole point of exhaustion on surviving death's door is to punish reckless and unrealistic play.
You need to ask yourself: How do I want to challenge my players? Do you want them to develop better tactics? To develop means of avoiding combat instead of rushing in headlong all the time? Are they taking chances to avoid combat and save time and prevent harm to themselves? Are you awarding equal XP for the PCs doing that? Do you want your players to develop better strategies for how they approach an adventure's goal, or are you satisfied with letting them always choosing the sword to solve their problems, and just relaxing the consequences?