Yes: a non-magic Barbarian that focuses on support is in XGtE
I am going to ignore the least useful part of your question in offering this answer.
Is such a build possible which is at least roughly as useful as other, more typical builds? And would it remain viable at higher levels?
Viable in particular as a criterion is a red herring in D&D 5e, given bounded accuracy, given that this is a team based game at its root, and given that you provided zero measurable criteria from which to discern viability. What follows is a useful support Barbarian build, from Xanathar's Guide to Everything, in the form of the Ancestral Guardian Barbarian. (XGtE, p. 9-10)
There is an inherent contradiction in your problem definition, but we can work with it
a useful combatant who doesn't deal damage (at least not primarily)
While that criterion is nebuluous it gives us something to work with since the edges aren't well defined. In a party with at least one other martial character, and better with two, the Ancestral Guardian Barbarian support character can be very useful.
At entry levels
Since you do not forbid combat, but are not emphasizing DPR for this character, at levels 1 and 2 make sure that the barbarian has javelins or hand axes, which have the thrown property, so that our barbarian can attack, trigger rage, and then be in a position to shove/grapple various opponents so that the other PC's can kill them.
Once in a rage, this barbarian enters into melee with the intention to grapple (or on certain occasions shove or knock prone) their opponents. This holds the enemy in place while the other martial characters, and even spell casters, do damage to them. With rage's resistance to melee attack forms, this support character can wield a shield, use the "free hand" to grapple, and keep the enemy from fleeing while not taking too much damage since such melee damage as is received is halved. (Slashing, bludgeoning, piercing) Now and again a "help" will be a more useful way to help your allies lay the lumber on the enemy.
Stat and PC Race advice:
Go seriously min-max on this: Con>Str>Dex>Wis>Cha>Int.
While you need some Dexterity for increasing your unarmored armor class, and it's a commonly needed saving throw stat, I'd boost Constitution first, and then bump up Dex. The Barbarians second level feature (Danger Sense) mitigates some of the cases where you rely on Dexterity by giving you advantage versus traps and spells that you can see and that require Dexterity saves.
Mountain Dwarf won't be a bad racial pick, due to resistance to poison and advantage on saving throws. A variety of monsters throughout the game do poison damage. With +2 to both Str and Con at level 1, your two most important stats are already blessed with bonuses.
In almost any situation, this Barbarian can "help" rather than attack and provide other melee attackers with advantage on their attacks. While so doing, the initial rage may expire if you have not taken damage, but the other party members should have been able to build some momentum for that encounter.
At level 3 Ancestral Protectors.
The point of this feature is to, when you rage, make all attacks on others be attempted at disadvantage by the creature that you have chosen. You still have to attack (javelin, ax, battle ax) and you have to keep the rage going for this support skill to stay alive, but this makes all of your other front line characters less likely to be hit. You de-buff a selected enemy. (Pick the hardest hitting one, eh? You go after the ogre, not the goblins).
Pick Con for the ASI at 4 (or the Grappler Feat)
You want to benefit from Unarmored Defense, and get more HP, and make your Con saves. (See below)
Embrace the second attack at level 5
Make an attack with one (to keep your rages up) and grapple with the other. You hold the enemy in place, and your allies kill it. Rinse and repeat. You'll lay a little lumber while you are at it.
Sixth level gives you Spirit Shield
One of your allies that you can see gets 2d6 damage reduction. This goes up to 3d6 and 4d6 at 10 and 14th level. You have to be raging, so you still have to be either attacking or taking damage, but this is a very nice support function for your party. Someone else takes much less damage: support function.
Consider Grappler Feat at 8 (If you didn't take it at 4)
You'll get advantage on attacks of grappled opponents. You can make a second attempt to restrain the creature, so that your allies get better attacks/damage against them.
Possible 1 level of multi-class in Rogue
Get athletics Expertise. Get another skill. Your grappling just got better. Your shoves got better. Your knocking prone got better. If you knock an enemy prone, and your paladin ally attacks twice with advantage, Divine Smites can get pretty nice nova damage, particularly on a critical hit.
- @mattdm suggests (if the build is going to 20) consider fitting in three levels of Rogue and take the archetype Mastermind: this provides Help as a bonus action and at range. This will depend on your party's composition, but might be a good it.
Vengeful Ancestors
This adds damage of a hard to resist type, force, but at level 14/15 when you get it, your allies have a lot of different ways to exploit the support you are already offering.
But sometimes the best defense is a good offense
Now and again, you need to be supportive by whittling down that big bag of HP that a given monster has. So let's take a look at generic Barbarian Features.
- Reckless attack helps with that, above and beyond support.
- Extra Attack helps with that;
- Brutal Critical helps with that;
- Relentless Rage helps in keeping you up, keeping that rage bonus damage going, and supporting round after round, until you miss that Constitution save (See above for why ASI in Constitution is a good thing)
- Persistent Rage aids in the entire Ancestral Guardian approach, since rage fires those abilities.
- Indomitable Might: better grappling, better shoving, better Help actions ...
- Primal Champion (If you don't choose that one level multi-class in Rogue): Strength and Constitution go up by 4. All of what you are doing above is improved.
