This seems like it would greatly slow down play if you keep the standard that a long rest only removes one level of exhaustion.
It all really depends on the frequency of the encounters you throw at your players. For example, I'm running OoTA with this rule, and since they run into trouble in the underdark wilderness only every two or three days ( two times a day if they're particularly rash or unlucky), it hardly slows down the rythm of play.
My intent with those rules is to make them think before engaging in combat or taking big risks. I feel like combat isn't something you should recover so easily from. With the clean slate long rest, I have to constantly throw bigger monsters at them, or build my campaign around six encounters/day. Wich is a bit absurd.
Mostly, when they get into a fight that's more difficult than anticipated, and one or two of them got down, they spend a few minutes discussing if they should take a few days break or keep going. If they keep going, they get extra cautious, and I can keep them on edge with a few die rolls behind my screen. If they stop to rest, and take extra steps to ensure their camp's safety, I use the elipse method, and tell them a few days passed, and they got on their feet. ( except, of course, if they didn't take the necessary precautions, or if I had other plans for them.) Hardly slows down the game and gives them a feel that things are dangerous and exciting. It gives me more options to keep them excited.