The problem still remains that such situation are ones created to make it useful and are not very common as the party needs to be resting outside where there is bad weather that the gm pretty much declares is a problem.
So one thing we have to keep in mind, which is why white room analysis can only take us so far, is that the GM has a massive (I would say dominant) impact on how balanced things are in an individual game. Part of the DMs job is to highlight character abilities one moment, and highlight their weaknesses the next.
If I am playing a fireball focused sorcerer, and the DM uses lots of encounters with like 20+ guys....I will seem ridiculously embarrassingly overpowered compared to the rest of the group. and if the DM uses nothing but single big monsters, my guy will seem like a joke. A big factor of 5e is the assumption that the DM will utilize the "hooks" of the various classes to make them look cool.
This is why when I see people argue that "Monk speed is useless", that means they have never had a DM that used that hook. I guarantee you have a few fights where the McGuffin to turn off the heinous XYZ is 100 feet away from the group, and suddenly the 50 ft speed monk is going to feel really cool.
My point for this debate is that...yes, a DM can absolutely make Wizards stronger or weaker by how they interpret spells or how they rule effects. A DM that kills a familiar the second it sticks its head anywhere is going to make a wizard weaker than one that lets the familiar by a perma nigh invisible spy that all monsters ignore.
But the key is....Wizards have LOTS of hooks that can be used for cinematic effect. Divinations give them the chance to dictate questions and engage with the DM. Familiars and unseen servants lets them "handle the action" while the party maintains safety. Spells like Leomund's hut let them dictate where the party rests in the bad weather (which a DM who knows their wizard like leomund's hut is going to put out every so often to make that character feel good, because that's what good DMs do).
Now back to combat for a moment. I have seen a number of fighters and wizards in my game. The fighters are solid in combat, they really are. Wizards are not the dominant powerhouses they once were. But Wizards are still
cinematically very strong
. One example that happened in my game, party encountered two iron golems. Party wizard simply snapped his fingers, used a 5th level banishment on both of them. No one cares about magic resistance when your charisma save is at a -5. Both golems defeated in seconds, concentration doesn't matter when all the enemies are gone. Now could the party have fought the golems and won? Probably, it was a tough encounter (my group was 12th), but I think quite doable. But cinematically, the Wizard looked just insanely awesome, I mean people just stood back and gasped at that moment.
Fighters don't have the same tools to create those awe-inspiring moments. Doesn't mean they aren't able to, again 5e expects players to get creative and not just use their mechanics. For example, in my 20th level campaign our barbarian was facing a Lich, a mind flayer, and a rakshasa. He took the parties staff of power, ran into the lot of them, and broke the staff. Kill the 3 of them instantly, and was tough enough to survive the explosion. Bad ass moment. The difference is that wizard come embedded with a lot of hooks, so its relatively easy to generate those moments, fighters have to think out the box a bit more.
I can also say as the DM, the casters occupy a lot of "thoughts". Most of my combats and encounters I have to design to account for caster abilities more than fighter ones.
And now to bring it all back to the actual OP of this thread and actually get us back on track for pete's sake. Long Rest vs Short rest, a good DM can absolutely balance both types of classes. Here's the issue though.... as a DM
I don't want to. I don't want to have to create 6 encounter days every freakin day in order to have balance. Can I...of course, but that's not how my campaigns are designed, and I have 0 desire to change my campaign structure to fit a balance need. So in that respect I do think the long vs short rest dynamic has failed, and I think it better that all classes are balanced around a similar chassis.