I think it's even fine to do a party limited to only the Barbarian (Berserker), Fighter (Champion), Fighter (Battle Master), Rogue (Thief), and Rogue (Assassin). It's a narrow suite of options, but a party made of those five characters would, I believe, handle the challenges 5e throws at it just fine.
It could handle straightforward combat challenges, as long as they face no set-backs, and whatever challenges can be answered by a high-DC skill check in whichever skills the Rogues have Expertise in.
A combination of fast combat, an emphasis on managing the context for an encounter (ambushes, scouts, etc.), and the suite of combat options available to all characters make this a very viable party.
It could be effective for a while: it has lots of DPR and some exploration skills. So it could prowl around, getting surprise, winning quick victories, and evading contact once out of HD - for a while. But they couldn't cope well with even seemingly small setbacks, like being surprised instead of gaining surprised, or a few good rolls dropping one or two of them unexpectedly and a fight dragging out as a result. So not really viable. The DM could avoid those sorts of things by narrowoing the scope of the campaign and nature of the challenges enough, of course, that's always an option, no matter the party.
the common availability of Healing Potions,
Obviously not applicable in a no-magic game.
I think you under-estimate the "builds" of 5e.
There just not a lot of customizeability or flexibility in the 5 PH non-magical sub-classes ('builds'). You have two very combat-focused tough multi-attack-DPR 'builds,' two opportunisti-SA-DPR skill specialist builds, and one daily-Rage-DPR tough slightly-wildernessy 'build.'
I also think you put too much emphasis on roles. There is no critical need in 5e for a class or a character to be a "leader" or a "controller" or a "defender" (or a "striker!"). Party balance doesn't rely on having individual characters dedicated to these positions.
Sure, 'Role' and 'build' are defunct terms, and party 'balance' is subtler/more-DM-involved, maybe even as much art as science. But there are contributions that every party needs, and they're spread out over the classes. DPR is one of those contributions. Exceptional checks in a skill specialty or few are another. And those are all the significant contributions the few non-magical sub-classes have to offer. But, there are more contributions needed, more than just the absence of the other 3 non-Striker 'Roles' would imply.
The fact that a Champion Fighter's class features help her deal buttloads of damage in no way stops that Champion Fighter from doing other things.
The inability to do other things well enough to be useful or viable doing them is what stops her. In theory, that's some sort of balance or differentiation. You can hit things really hard and stack up crazy DPR, therefore you don't have to be able to do anything else, others will cover those functions.
Which isn't to say that more diversity wouldn't be appreciated, just that it's not necessary. There's nothing 5e is missing when it comes to a non-magical party, nothing that prevents it from realizing that reality. That adequacy is just that - adequacy. There's plenty of room to dive more deeply. But there's nothing missing.
There is obviously a great deal missing when it comes to non-magical options. Non-magical classes, for instance. All the important functions a party needs apart from DPR and Expertise. In-play flexibility. Mechanical coverage of non-caster character concepts (RP coverage being trivial, you can RP any concept, even if the game offered only one class). Whether what's missing is 'necessary' or what little there is 'adequate,' can be chalked up to opinion or PoV or context.
But, I think we can agree that 4 non-magical classes is more than 0, 8 builds out of 18 more than 5 out of 38, and 3 out of 4 formal roles more coverage than 2 out of 6* or 9 or however many sorts of definable contributions may be present/needful in 5e.
The fact that 5e manages to allow for any character to adequately cover any combat role in a typical 5e combat (short as they are) is part of why I think 5e does this better. Rather than specifying that it's Class X's job to heal and Class Y's job to deal damage, they leave the decision as a tactical one for the individual player on each of the PC's turns - does your party need healing? Do you need to take out an enemy quick? Do you need to protect a vulnerable party member? How do you do that in the moment? Did you come prepared? Can you risk doing something you're maybe not the best at for a round or two?
More than once in a 5e game, I've uttered the term "Tankthief," when a character with d6 HD who hasn't been hit as much as the main melee machines takes over for a round or two on one of the fronts.
Y'mean the d8 HD Rogue when you say 'thief?' Actually, it's funny that one thing 5e did map almost precisely from 4e was the relative hp/level of the classes d10/6, d8/5, d6/4. CON bonuses throw it off completely, but the base is there, no more d4 HD classes, rogue & bard 'promoted' to d8.
5e is set up well to allow moments like that to happen. If that happened in a 4e game, it would've been bad news, a sign of the defenders or controllers not "doing their job,"
No, 4e did not assume that having a Defender meant no one else ever got attacked. In fact, if that happened, they were 'doing their job too well,' since other classes did have hps & Surge resources for a reason. Not that 5e's relative lack of 'defender' mechanics is all that relevant: the all-non-magical PH-only party doesn't lack defenderish capacity to any greater degree than the party pulling from the whole PH, nor even the all-magical PH-only part.
But I think it would be missing some of 5e's most remarkable design elements to characterize the out-of-the-box capability of 5e doing a non-magical party as inadequate. It's rather amazing what putting healing potions on the equipment list can do in a world of 3-round combats.
If you need healing potions to pull of a no-magic setting, the system has failed to support that style of campaign. And if the game can only handle 3-round combats, it's failed to support a range of playstyles. I'll accept that 5e hasn't yet succeeded in the former, but hold out hope that it's only a matter of time. The latter, however, I don't think should be the case, 5e is heavily tuned towards fast combat, but the DM is sufficiently empowered to construct and manage the flow of more challenging combats to allow them to go more than 3 rounds without the whole thing completely falling apart (though, doing so without in-combat healing available to the party would be too much to ask).
Here's a thought experiment on 'adequacy.'
If the PH had an equal number of magic- and non-magic-using sub-classes, would the magical side be 'adequate.'
Let's get even more specific. To match the 5 non-magical sub-classes (Berserker, Champion, Battlemaster, Assassin & Thief), we'll posit these 5 magical sub-classes: Totem Barbarian, Eldtrich Knight, Arcane Trickster, Hunter & Beastmaster.
That 'adequately' cover everything you'd like to do with magical PCs?
