We aren't talking about second & third edition* with the 5e hit dice mechanic grafted on. The HP attrition model present in 5e is not one where going into the next fight with all most or only a few HP is going to be a meaningful concern due to PHB197 & healing word/1hpLoH/Healing Light/etc.Or you need the short rest to restore fewer resources.
The core short rest mechanic is spending hit dice to regain hit points. Since your hit dice don't recharge until a long rest, more short rests would not shift the balance significantly. (It would slightly increase the value of hit dice, because spreading your hit die rolls out across more short rests means less "overkill" where you roll higher than expected and hit your max. But the benefit of this would be very small.)
The place where balance concerns arise is class abilities that recharge on a short rest, and in a lot of cases (e.g., monk ki/discipline/whatever they end up calling it) you can simply chop the ability's points or uses in half and call it a day. I only see two class features that would require serious effort to rebalance: Warlock spell slots, and fighter Action Surge. Spell slots are the toughest because, with the right mix of feats and multiclassing, they can be used for almost any spell in the book, which opens up a vast number of possibilities for abuse. Action Surge is difficult because it's unitary -- you either have it or you don't -- and very powerful.
Still, these are not unsolvable problems.
*maybe 4e too, I dunno