What else would be needed for a no-magic game that can't be compensated for by feats and other available choices?
Feats aren't necessarily available, but, that aside, there's the whole range of dynamics that make D&D challenges work, encounters, particularly.
As I pointed out before, a DM who decides to run a no-magic game isn't going to throw challenges that require magic at the party. Large AoE control or damage effects, reliable in-combat healing, flight or teleportation etc are all the province of magic, and thus their absence will be compensated for.
You're painting your way towards a tautology, here. Sure, if you assume that the DM will compensate for anything the party lacks, it doesn't matter what the party has going for it. You can posit a party who all do origami, and solve all their problems that way, because you assume the DM engineers all challenges so that origami works.
It's really not necessary to go there. Take flight, for instance. In a no-magic setting, no one's casting Fly, and you're not going to be facing creatures that levitate or live in cloud castles or anything. There might still be improbably large winged creatures flying around, so there may be enemies able to fly or reach places very difficult to reach without flight - but there will also be, well, large winged creatures that you ride. Similarly, teleportation - if no one else teleports, there won't be a lot of call to reach places that can only be reached by teleportation. There are problems that exist only because of magic, so not having magic to solve them because the setting lacks magic wouldn't even theoretically be an issue.
OTOH, three are problems that aren't magical in nature, that, in D&D, are still, by far, optimally solved by magic, even though, in genre, they are often solved by other means. AE damage lets a party blow through a lot of enemies efficiently. The party may still need to do that - there are enemies out there, no reason there couldn't be a lot of them, and heroes in genre do things like that sometimes. Extra attack can grind through groups of enemies, though not as efficiently as it can beat down one at a time, and SA can't. (In past Editions, Cleave, WWA, and Close attacks made that sort of things more practical. 5e has virtually nothing along those lines, but such options could be easily added.) Or in-combat 'healing' - D&D combat has always relied on PCs being able to get hps back in the middle of a fight, without that capability, there's too little margin for error, especially when PCs start dropping. (Non-magical in-combat healing was quite adequate in one past edition, it's not absent in 5e, but its inadequate - the game just needs more of it.)
Then you have battlefield control (3.5 fighter builds could generate a lot of it, 5e, not so much), single-target control (conditions, 'defender' mechanics, which, again could be done - barely adequately - in past editions without magic, but which 5e doesn't deliver to the same level, and should really exceed what past editions did to bring things up to where they 'should' be, both in terms of maintaining playability, and in evoking genre), multi-target control (even more so, something that's at best been nearly-adequate in the past, and which 5e has the flexibility and design space to do better), buffing, damage mitigation, condition mitigation, mobility, etc, etc... 5e at least has a tiny bit of each, establishing that they can be done without magic, it just needs to expand the range of choices and flexibility available to PCs in such campaigns to a workable level.
You appear to have a very specific and personal image of what a Druid is, and what they should look like.
The D&D Druid does have some basis in myth, fiction, and even history, and those sources don't exactly paint it as consistently a non-combatant. As with all casters, D&D tends to give it too great a range of abilities, to readily available to each individual member of the class, relative to what's actually displayed by individual characters from those sources of inspiration. Oddly, considering what Hussar's complaining about, it's not usually being able to pull one magical trick repeatedly that the problem, in that sense. If a Druid in fiction can conjure fire or change into an animal, he can probably do it quite a lot, but probably can't do a whole lot else without elaborate rituals, divine intervention, and/or special materials. It's the large number of know spells, length of class spell lists and the oddity of Vancian daily 'slots,' that make magic seem so casual and ubiquitous in D&D.
Well, on one level, it's simply a preference, and doesn't really need to be justified. Some folks want more in a non-magical wilderness-oriented warrior than they can wring out of a Fighter with the Outlander background.
But to get to the core of why, the answer is probably simply that 5e, in trying to take the best from all prior editions, and capture the feel of the classic game, in particular, has gone back to mapping specific mechanics to specific fluff. Spells aren't just bundles of mechanics that can readily be re-defined to represent psychic powers, extraordinary skill, uncanny knacks, or outrageous luck - their magical nature is woven into the way they're described, what they do, how they do it, what it takes from the user to make them work, and how the game resolves them.
What is the distinction between a magical ability and a spell or spell effect that causes problems?
Between those two, not a huge amount - slots, mostly, I'd guess - oh, and casting & components, too.
Spell slots, Ki points, Superiority dice etc are all resources that can be spent as the player chooses on abilities from a set list, picking something that fits their specific situation. Its more fun, and better from a game design point of view than having a long list of x-per-day abilities, many of which may well rarely or never be used.
I have to agree with that. It's also expedient to have any non-spell abilities still use some sort of very flexible resource, so as to make them more nearly comparable when evaluating classes. And, it's also better for classes to have such abilities instead of depending entirely on unlimited-use contributions to the party, even if they are only locked-in & x/day.