There is some merit in having high ability scores other than, say, Str/Dex and Con, but you get less benefit from a 20 than you do from proficiency by endgame. Bulking up Wisdom and Charisma for better skill checks and being able to save against the effects that matter (to a lesser extent, Intelligence) is where classes like the Barbarian and the Fighter, especially, need the most help.
By Tier 4, if you have a Wis save of +1 or +2, you're toast. You need to get Resilient for that save. You're still going to be absolutely nuked by Charisma saves, but at least those are less common. I started to see this trend in Tier 3, so I can't even imagine what a Tier 4 game without the DM being very selective with foes, or casters to mitigate status conditions is like (heroes' feast alone can turn a TPK against a dragon into a winnable combat).
What your non-casters need, really, is more ways to get proficiencies or advantage on rolls as they level up. It's telling that the "masters of warfare", the Fighter, can't even get reliable advantage on attack rolls, something Barbarians get for most of their career.
And Indomitable really needs to be better. At the level you get it, you can reroll a save, but the DC might be nearly impossible to hit, depending on what it is- once per day. A level after this, a Monk gains immunities. At 13th level, the Fighter can use Indomitable twice a day. One level later, the Monk gains proficiency in all saves, and can spend a ki point to get the same benefit as Indomitable!
I'm not saying Monks are better than Fighters, they have their own problems, but still, one of these things is not remotely like the other!
And even if this covers the numerical weaknesses of the class, WotC has deliberately designed narrative power to be skills > spells, and not just in a minor way. Sure, you can use skills more often than spells, but spells can do way more than what a skill check can accomplish. And it's not like casters don't get skills either! Plus they get things like guidance as a friggin' cantrip to boost chances of success- what at-will methods of increasing chances of success do other classes get that aren't wholly situational?*
A real problem is that Backgrounds, which should be able to obviate or buff skill checks a lot, lack any mechanical "bite" and get ignored in a lot of games (maybe not your game, but it happens), which is why WotC is now looking to overhaul them completely.
*Other than the Rogue, obviously.