No, Rage is only replaced when you enter the Supernatural Fury

The way this is worded, the Rage feature is not removed upon reaching level 3. Instead the feature is disabled while in the Supernatural Fury. This means that you could enter a "regular rage" but these features are disabled while in the alternate state.

This shows that the wording is probably flawed

Because of the earlier bit in the feature: "You gain the ability to cast and concentrate on spells, even while raging", you could also cast spells while in a "regular rage." This is almost certainly not the intent.

Take the Mobile feat

Your goal should be to get into melee, apply your ancestors to the boss, and get away. With the Mobile feat (PHB, p. 168), you won't provoke opportunity attacks from enemies you attack, and you'll have extra movement (in addition to a barbarian's own extra movement) to leave range. If at least one of your allies stays near the enemy, the enemy has a tough decision to make: eat an opportunity attack (or three) or attack your now very tanky allies.

In practice

One of my current Adventurers League characters is an Ancestral Guardian / Battle Master using this strategy. When I originally wrote this answer, he was an Ancestral Guardian 6 / Fighter 2. Since then, he has capped out at Ancestral Guardian 16 / Battle Master 4.

Throughout all tiers of play, he has been extremely effective at preventing damage on myself and my allies. Why be a Bear Totem barbarian when your ancestors turn your whole party into Bear Totem barbarians! As far as encounters and magic items go, AL tends to have 3-4 encounters per module, and my character only had a +1 weapon for levels 5-7 without feeling weak.

Other considerations

• You mention that the Champion fighter in your group is cowardly. The nice thing about this build is that (unlike the Cavalier from Xanathar's) you can...encourage...allies to synergize with you without forcing them. Fighter not tanking? Apply your ancestors and use your 50' of movement to run past your cowardly ally. The enemy will likely engage with the fighter since it can't reach you. When it misses the fighter, play up how your ancestors distracted the enemy. Your allies will learn that ancestors equal safety.
• It's usually better to apply your ancestors to the least damaged enemy first and then leave them so you can attack the most damaged enemy with the rest of your party. Your ancestors do nothing if the enemy is dead.
• If an enemy is highly mobile and dangerous, you can stand back and apply your ancestors from 150' away with a longbow.
• If you also multiclass into fighter, the Battle Master's Menacing and Goading Strikes (PHB, p. 73-74) are like a mini ancestors effect. They synergize well with the rest of the build.
• As @Ben Barden pointed out, the Sentinel feat on the fighter (PHB, p. 169-170) in your party would assist with this playstyle by preventing even mobile enemies like dragons from leaving to hit you.
• I would not recommend taking any feat or other feature that uses your reaction. At Level 6, you get Spirit Shield, which prevents an average of 7 damage per round as a reaction. That's similar to casting healing word every round without using resources! With the hit and run playstyle, you often won't be next to enemies to use an opportunity attack, so you can (and should) use Spirit Shield every round.

Bonus math

This build always feels strong in play, but I was wondering how strong it actually is.

TL;DR: the wizard, protected by ancestors, survives longer than the reckless-attacking barbarian.

The PCs

Since this question focuses on survivability, I am assuming that the barbarian is using a one-handed weapon and a shield. If not, a hit-and-run playstyle becomes even better. (I would recommend a two-handed weapon to anyone trying this build.)

The monsters

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).

The ancestors

I will assume that the barbarian's ancestors are applied to the Gladiator every round. (There is a small chance that they will miss both attacks, even if they attack recklessly for advantage.) For anyone but the barbarian, the ancestors impose disadvantage on the Gladiator's attacks and will provide resistance against any damage the Gladiator may do.

The numbers

As a baseline, the barbarian will last 4.4 rounds against the Gladiator while attacking recklessly, or 6.9 rounds normally. The barbarian numbers assume that they have resistance to the attacker's damage. If the party faced a CR 5 Fire Elemental instead of the Gladiator, the barbarian no longer has resistance, but their ancestors still protect the party against all attack damage.

The wizard can last around 5 rounds protected by ancestors if they cast shield on the first two turns of the fight. If the barbarian took the Tough feat (gaining +2HP per level) instead of Mobile, they would last 5.2 rounds recklessly, or 8.2 rounds normally. These are still a far cry from the 18.4 rounds the fighter would survive.

The conclusion

The fighter becomes a truly incredible tank, surviving more than 5 times longer than without the ancestors. In fact, every PC survives longer than a barbarian who is attacking recklessly (all while allowing the barbarian to do so without penalty by keeping out of range). Who knows, you might even make a brave warrior out of your cowardly fighter ally!