Frame challenge: 1 ability in the hand is worth 2 in the bush
This mostly came from a comment I made but I have seen questions from people with a similar mentality to my own asking how to avoid it, so I know I am not alone here.
But to me PB times per long rest is something that I feel I have less control over than X times per short rest.
With long rest features I am far more likely to save them for later, sometimes ending the day with several uses left.
With short rest features, I can (somewhat) control or predict when I am going to be able to rest. It matters less how many rests I will get, only how many fights it will be before I get that rest, and with things like tiny hut and rope trick short rests are very easy to obtain.
So PB may be more times, but if you don't use an ability then it is worthless.
That said, unless you use constant time pressure, short rest can easily be more
As mentioned above, a short rest can be made almost entirely safe with a rope trick (single action cast) or a tiny hut (10 minutes cast so a bit more DM discretion) and can be done almost any time. Many people don't play with constant short rests because the standard day at a lot of tables tends to be only a few fights, but players have much more control over short rests, so effectively can often have as many as they need. If that doesn't make short rest powers used more often than PB it is because they don't need to use them that much, so it makes no difference anyway.
To answer the premise of the question:
The reason you asked was to evaluate both options for balance, and while this is opinion, it is educated opinion; I would count them as equal.
Generally PB is likely more, but short rest has enough benefits that it comes out either even in terms of value, or very close to it in my eyes.
It is also telling that a lot of formerly short rest powers have been changed to PB with little actual change, which tells me they can change to the latest flavour while still being balanced.