I've used gritty realism extensively. It is probably my favorite "switch" to use in 5e. It makes the game run a lot better, and it's an easy "fix" to a lot of problems that my table had. However, given that a lot of 5e has a lot of default rules and assumptions baked in, there isn't just one simple fix ("When all you have is a hammer, every problem looks like a nail"). If this is the only rule you tweak, you end up seeing the following:
A. As noted by the OP, Fighters and Rogues (the "always on") and, to a lesser extent, Barbarians really come into their own. It makes martial characters more interesting, and makes skills (and the use of skills) more valuable in the game.
B. Unfortunately, it pretty much destroys the monk. The monk is the only martial that truly depends on a resource (ki) that is replenished during a short rest; using this variant without more rule tweaks turns the monk from an interesting martial choice in a campaign to a nearly useless one compared to other characters. While the other benefits of the monk (such as unarmed and unarmored fighting) might appear to make up for this in a survivalist campaign, having a class that is so dependent on short-rest abilities for their regular martial abilities just doesn't work unless you make further alterations.
C. Full spellcasters become more interesting, albeit weaker. Players have to adapt to spell scarcity, and I have found that scarce resources become more valuable. There are a lot more interesting uses of utility cantrips, and higher-level spell casting is usually impactful and well-considered.
D. OTOH, half-casters and third casters .... not quite as great. Paladins aren't going around smiting everything every turn. I highly recommend using the variant (spell-less) Ranger.
E. Generally, though, this is the usually the beginning of a cascade of changes and other house rules. For example, with gritty realism, standard 5e-style recharging wands are really powerful, so you probably want to default to non-recharging wands. A feat like magic initiate suddenly becomes too much, so our table banned it (in a world where cantrips are overvalued, it's best to restrict cantrips to the actual spellcasters). And so on.
Gritty realism is awesome, but it's usually the beginning of the tweaks, not the end.