Under the standard 5e 'day' - 6-8 encoutners, 2-3 short rests, a 1/short rest ability could be used up to 4 times, but never more than once in a given encounter. There's an argument that a 4th use, sometimes, makes up for the 'Nova' potential and flexibility advantage of 3/day. I don't find it a convincing argument, but it could be made.
In games with shorter 'days' - that is 2 or fewer short rests on average - 1/short rest is strictly inferior to 3/long rest. It gives the same or fewer total number of uses, and has less flexibility.
Obviously, on the other extreme, 1/short rest is strictly superior to 1/day.
A more nuanced consideration is that a 1/n resource is particularly tricky to manage, as you want to use it at the optimal moment, but doing so precludes using it again for n time. So it's very easy to mismanage such a resource and decline to use it at the optimal time, and end up using it sub-optimally, later, or not using it at all. Conversely, it's frustrating to use such a resource, then find yourself in a situation where it would be optimal or even turns out to be absolutely vital. It doesn't matter if n is short rest or long rest or day or whatever. But, even 2/n becomes /much easier/, because you have the flexibility to jump in and use it in the first optimal-seeming opportunity, while still keeping one use in reserve. In addition, 2+/n means you can 'Nova' by blowing the resource completely on a single situation.
Bottom line, to 'balance' even theoretically, 1/short rest probably roughly equates to 2/day, under the 2-3 short rest/day assumption. The 1/rest ability can generally be used more times per day, and that helps make up for the easier management and nova power of the 2/day ability.