Your self-answer covers the standard approach well, so I'll take a stab at an alternate approach.
Or: How I Learned to Stop Worrying and Love My Mom
Picture this: three frost giants bear down on a party of adventurers. The fighter chops away at the closest one's ankles while the rogue threads a shot between the gaps in their armor. The wizard wiggles her fingers and the giants' eyes glaze over. One swings its axe and barely misses its ally, another stares dumbly at its hands, and the final one flees. The fighter and rogue complain that the wizard should have cast fireball and wish that they had just brought along another rogue.
Wait. That's not what they would say. Without inflicting a single point of damage, the wizard completely changed the course of the fight. Damage is life, control is life insurance.
The Control Barbarian
Most Barbarian Paths focus on increasing personal defense (Bear Totem) or personal offense (Berserker). One, however, focuses on party defense: Ancestral Guardian. At Level 3, they receive the following feature (XGtE, p. 10; emphasis mine):
Starting when you choose this path at 3rd level, spectral warriors appear when you enter your rage. While you're raging, the first creature you hit with an attack on your turn becomes the target of the warriors, which hinder its attacks. Until the start of your next turn, that target has disadvantage on any attack roll that isn't against you, and when the target hits a creature other than you with an attack, that creature has resistance to the damage dealt by the attack. The effect on the target ends early if your rage ends.
Combat as a control-focused Ancestral Guardian starts out like most barbarians. You rage, run at the biggest monster on the battlefield, and try to hit it in the face. From then on, your allies will scratch their heads and wonder if their barbarian is broken.
Once you hit your target, they will be harassed by a wide array of terrifying creatures; your not-mad-just-disappointed mother, way-too-much-perfume aunt, and have-I-told-you-about-my-time-in-the-war grandfather, will unite to tell your enemy why they should attack you instead of your allies.
Now that your biggest opponent wants you deader than usual, it's time to beat a hasty retreat. The Mobile feat (PHB, p. 168) lets you leave their reach without provoking an opportunity attack. Use it to stroll on over (with +10 speed from Mobile and +10 speed from barbarian's Fast Movement) to some of your smaller enemies and remind them that you're still a barbarian.
Your other allies can now engage the target of your ancestors with near impunity. Even if the harassed creature manages to hit with disadvantage, your allies have resistance to its damage. Congratulations, you just turned your whole party into Bear Totem barbarians who took the Dodge action. If the creature instead decides to chase you (assuming it can compete with your 50' speed), your allies get free opportunity attacks.
The Melee Build
- Basic setup: Rapier and shield. Half-plate brings you up to AC 19, but you have AC 18 in your skivvies, so you might not need it.
- Initial stats: 16 DEX and 16 CON are the only requirements. I'd recommend 12 STR, so you're still a decent grappler with advantage from rage. You could technically dump it, though, if you want.
- Level 3: You get Ancestral Protectors. Without the Mobile feat, this isn't quite awesome yet. You can use a whip's reach to approximate the feat.
- Level 4: Build online. Grab the Mobile feat and run around stabbing anything that moves.
- Level 6: You get Spirit Shield, which prevents even more damage without using any resources.
- Level 8: +2 DEX. Your unarmored AC is now 17+2, so you can ditch the half-plate.
- Level 12: +2 DEX. Your unarmored AC now equals plate!
The Ranged Build
Ben makes a good point in the comments - since Ancestral Protectors works at range, you can make an entirely ranged character if you want.
Instead of a rapier and shield, you just need one hand crossbow. Grab the Crossbow Expert feat (PHB, p. 165) in place of Mobile to get an additional attack (except on rounds when you begin raging). That feat also removes the disadvantage from ranged attacks while you're in melee, allowing you to still run up and tank. Later on, you can grab the Sharpshooter feat (PHB, p. 170), particularly if your party is good at generating advantage. The question requested no multiclassing, but optimally you would dip a level of fighter for the Archery fighting style. The only thing you really lose is your shield (since you need a free hand to reload the crossbow); in exchange, you gain an additional attack on most combat turns. If you need really long range, you can easily pull out a longbow.
One of my current Adventurers League characters is an Ancestral Guardian 7 / Fighter 3 using a strength version of this build. He has been quite fun to play and is extremely effective at locking down big hitters.
(This is outside the scope of the question, but don't be afraid to make a strength-based Shadar-kai. Guess what the strength score of my strength-based 10th-level barbarian is? Did you guess 16? Don't tell anyone, but I never raised it!)
How effective is he, really? The short answer is the most convincing: a wizard, protected by ancestors, survives longer than the reckless-attacking barbarian. (You obviously won't be using Reckless Attack, but the numbers are still impressive.) What follows is a condensed version of the math from my answer to "Most effective way of improving survivability for an Ancestral Guardian Barbarian".
| CLASS | AC | CON | HP |
| Ancestral Guardian 5 | 19 | +3 | 55 |
| Champion Fighter 5 | 20 | +3 | 49 |
| Hunter Ranger 3 / Rogue 2 | 16 | +2 | 42 |
| Evocation Wizard 5 | 15 | +1 | 27 |
| Archfey Warlock 5 | 14 | +2 | 38 |
This fight will be a Deadly encounter, as it is the boss fight. The PCs will face:
- One CR 5 Gladiator (MM, p. 346)
- Five CR 1 Duergar (MM, p. 122)
A fireball from the evocation wizard will seriously injure or kill several Duergar, leaving the rest for cleanup by the warlock and the ranger/rogue. With that in mind, let's focus on the Gladiator. The Gladiator attacks three times with a +7 to hit, each dealing 11 damage on average (18 on a crit).
As a baseline, the barbarian will last 4.4 rounds against the Gladiator while attacking recklessly, or 6.9 rounds normally.
| CLASS | BASE ROUNDS | ANCESTORS |
| Champion Fighter 5 | 3.4 | 18.4 |
| Hunter Ranger 3 / Rogue 2 | 2.0 | 7.0 |
| Evocation Wizard 5 | 1.2 | 3.9 | No shield
| Archfey Warlock 5 | 1.6 | 4.7 |
The wizard can last around 5 rounds if they cast shield on the first two turns of the fight. The fighter survives more than 5 times longer than without ancestors!
Will your dexterity-based Ancestral Guardian barbarian top the DPS charts? Nope. Will your Ancestral Protectors feature draw envious looks from control wizards? Probably. Will you have a blast running laps around enemies while your mom nags them so hard they fail at their one job? Definitely.