I really do not see anything in the game even implying skills ought to be able to achieve anything approaching what casters do... in fact the opposite is true when the game explicitly tells Players in the Players Handbook that without magic adventuring would be ten times more difficult. And the paltry amount skills advance sheesh. And spell casters including Bards are as capable of being the ones who have that mundane solution plus have the spell casting in addition to that.
D&D just has this crazy milieu. My understanding of it all is that IN THE BEGINNING there was the fighter. The 'Fighting Man' was literally exactly, mechanically, the Chainmail 'veteran', a 1d6+1 hit die infantryman. Normal troops were 1d6 size hit die, and elites 1d6+1. So, in the fantasy supplement you had heroes (4 dice) and super heroes (8 dice IIRC), which were 'leader types'. Chainmail of course only covers combat, and leadership to an extent. Magic Users and IIRC a form of cleric, also exist in the fantasy supplement, with a few combat spells included.
Nothing was really said in the rules about abilities, beyond the 6 scores, which were only a relative measure (no objective scale was given) in the core rules. It was only explained that fighters of 9th level (lord, the highest level covered in the core rules) were 'barons' and could control territory and collect taxes. The point being, nothing was really there in the rules that made either wizards, or clerics, 'more powerful' than fighters. Clerics could heal, and sometimes turn or attack (and can fight OK) and wizards have a very limited set of attack spells, and harsh rules on when they can be used (basically you can't cast in a fight, any casting better be done from ambush or similar situations, or from behind a line of infantry).
That was the origins of the D&D fantasy milieu. So, while the fighter was meant to cover all the heroes of history, or most of them, it was left up to the creators of a given game (and the rules were very clear that you were supposed to provide this) all the 'other stuff' to make your fighters Cuchulain, Vainamoinen, Siegfried, etc. The fighter was told to be restricted mostly to 'magic weapons', but then a bunch of other 'fighter equipment' was provided, and it can be assumed if you wanted the game to really 'go fantastic' then you would provide even more fantastical abilities through 'equipment' (which could include any sort of magically induced ability, not just from loot).
Because the game was intended as a campaign, with many competitive players (and in the start it was mostly armies competing, not individual PCs, that was the premise of the 'Great Kingdom' of the LGTRS). So it wasn't deemed 'fair', or basically in the interests of that campaign structure, to make super-powered characters who would outshine armies. There also wasn't anything to do with them, there were no 'dungeons' at that stage, not until DA made Blackmoor, which was a riff on a Braunstein AND Chainmail Fantasy Supplement, with some Ironclads rules thrown in (armor class and separate to-hit rolls). It was only at that point, with the first real D&D that wizards and then clerics got in on the act and PCs 'got super' for real. Unfortunately the milieu became set, and hasn't changed since. Fighters were just leader types for armies and infantry, magic users and clerics became the big shots as the spell lists got expanded and they could now do a lot of stuff besides fireball and magic missle or 'cure light wounds' and 'resurrection'.
So, we're stuck with this silly milieu that really cannot emulate most of myth and legend, and a lot of bad excuses for it. Maybe class is the whole problem. How about if levels 1 and 2 don't HAVE a class, and you only get spells or whatever from packages. The first one can come at level 3, etc. Honestly, I like my game better where progression works in an 'inverted' fashion. When you get treasures (major boons, ones that give you powers) THEN you level (once per boon). What you ARE is basically determined by the boons you get, which are driven by the PC's choices in play. The idea of a problem like "you're a fighter and it is anathema for you to 'do magic'" is ridiculous in THAT milieu. Sadly 5e simply isn't equipped to support it. More sadly, the D&D community cannot wrap its head around anything but the stock milieu, who's origin and purpose and peculiarities seem opaque to most.