+ Proficiency/2 in untrained checks? Not really. Having proficiency is always better than Remarkable Athlete. Everyone gets some proficiencies.
Also, the high Constitution given by superior Ability Increase capacity and/or Athlete feat can make it.
If those ASI's don't go for feats, the fighter gets a +1 to one ability check advantage over the next guy, from levels 6 through 13. Most campaigns never reach 13, let alone 14, when the fighter's 6th-best feat/ASI tried to compete with a 7th level spell for usefulness.
The whole point of fighters in 5ed is to be a reliable, steady character, not subject to resource management.
That's the same as saying they're purpose is to be a baseline from which all other classes measure how superior they are.
It's closer to being wrong than it was in classic D&D, at least. In 5e, the fighter's DPR is the 'best' at many levels. 'Best' really meaning no one exceeds by a large amount or with any consistency. Other classes do use the fighter's DPR as a touchstone or baseline, but they mostly strive for it. They may not have to strive terribly hard, but they don't just start there and go up.
Outside of combat, OTOH, the fighter is the everyman baseline. He can't do much of anything everyone else can do just as well, and everyone else has some things they do better.
Sadly, that's not far from true in combat, outside of DPR, either.
I think that you are a little biased by 4th Edition,
That's silly. We've all been far more biased by the 33 years of D&D in which the fighter was strictly inferior, than by the 2 years that it was balanced before Essentials pushed it back into the beatstick pigeonhole.
The resolution of most out of combat situations is generally made by means of Skill Checks and careful roleplaying. You could also say that neither the wizard or the cleric have "unique" means outside combat (almost every spell that they could be casted or replicated too by a Bard, a Warlock or a Sorcerer... or even a Druid, a Ranger and an EK).
Lol. Yeah, a Warlock, Bard, Sorcerer & Druid might, among them be able to cast all the same non-combat spells as a Wizard, that makes the wizard 'not unique.' Tell us another one.
Absolutely everyone can make checks. Making a check does not make you 'unique' or exceptional, unless you can make it was a bonus most others can't touch. All the fighter can do outside of combat is make checks, the same checks everyone else can make. Below 13th level, he /might/ make 1/6th of the with a +1 higher bonus than if he hadn't been a fighter - but that probably won't translate to a higher, or even as high, a bonus as the other proficient PCs in the party.
With this argument, one can say that the Rogue (like the Fighter) can't really do anything "special" outside combat: anyone can hide, be stealthy or be acrobatic.
Yep, anyone and everyone can at least attempt all those things - they're just checks.
But, the Rogue can be significantly better at a few of them due to Expertise.
Same thing with the Fighter: he has more feats/ better scores than any other class.
Expertise. +2 to two proficiencies starting at 2nd level and rising to +6 with 4 proficiencies at high level. Those bonuses let the Rogue substantially out-do anyone with Expertise in the same proficiencies - at high level, even if they have a much better stat.
Fighter bonus ASIs: +1 to one ability check starting at 6th level, and +1 to a second starting at 14th, but neither of those +1's can actually make him /better/ than another character who has also maxed out the sat in question to 20.
Don't pretend to find those equivalent.
In risky situations (EG failing a SC to climb in a difficult mountain, moving against water courses, or whatever) he can use the Panic Button of Action Surge, a feat that a spell could not. The fighter has a better core, as the Rogue. Here resides his "uniqueness".
The situations you mention would be using the extra action to take another move - a Dash action - once or twice between rests. The Rogue can use cunning action to do the same, at-will.