Nothing in 5e's rules pins resting to the GM or gives control over them via the needs of "story". If anything the rules are actively hostile to the GM making any effort shy of "because I'm the GM fiat" to limit rests. I don't choose that word hostile by chance either, the rest mechanics have so many layers dedicated to ensuring that there is no bar too low in order to begin a rest and that any disruption shy of rocks fall can simply be ignored by taking another restI strongly disagree with the bolded statement. Class balance is not inherently built on short/long rests. They are built around class feats, abilities, and spells. The short/long rest is a separate mechanic that interacts with the powers. To determine this, let's ask if hit point balance is balanced around rest? It seems like it is, but it really isn't. Please let me explain.
PCs rest to gain hit points and hit dice. But during a battle, which is the primary focus on why PCs rest, hit points don't rely on any of this. Healing is so abundant during the game, especially during rests, that it is a very rare sight to have a whole group at half health walking into a battle. This is not a game where the wounded and limping warrior with an arrow to his knee and a broken arm hobbles to the next battle where only his will keeps him standing. This is a game where almost always, the party is at or near full health going into battle. And guess what, during that battle there is no interaction with short/long rests and hit points. Instead, time devices like bonus actions, actions, and reactions become the sole arbiters and mechanics to interact with hit points. And since combat is actually where your PC can die - this is the hp-mechanic that matters. (Especially at tables where the DM doesn't hide and lie to the players about rolls or a creature's hit points.)
This same thought can be applied to class feats, abilities, and spells. A short or long rest replenishes certain powers, but it is miniscule compared to the interaction of the story. The story, and thus, DM, dictate 99% of this. For example, your group decides to go into a temple of evil lizard folk cultists knowing full well their abilities to dispel magic and summon hundreds of lizard folk from the surrounding area. But the group really needs that swamp-staff that is inside if they are to stop a lich later in the story. So, they bravely go inside.
Now, here is where we pause. Unless you are a DM making sh*! up off the top of your head for the entire area, this temple probably already has a map and encounters in place. A really well-prepared DM might have a what happens chart, and therefore, they are able to explain to some bystander what happens if the alarm gong gets rung or what happens if the PCs try to parley with the lizard folk or what happens if they kill the queen inside. And if the DM is crazy prepared, they might have the surrounding area mapped out too: hiding places, trapped areas, spots where other creatures live, etc.
All of this is the DM's creation. All of it dictates encounters per day and the ability to have a short/long rest. And here is the other part that is just as important - this session looks as though the PCs will probably have a few encounters, maybe two, three, four, or five depending on their actions before they can rest. The next session might look completely different. There might be a single encounter and a chance to rest followed by another single encounter. It's all dictated by the story.
And here is what I will concede. If you are playing with a DM that is just making stuff up, yes, there is a good chance they fall into a pattern and repeat the same steps every. single. time. Encounter. Long rest. Encounter. Long rest. Encounter. Short rest. Encounter. Long rest. etc. This will cause some classes to far outshine others during combat. But again, changing the class and their interactions with short/long rests is not the way to fix this. It is a DM problem, not a class problem.