Trip ≠ Knockdown
The last column of Table 3-3: Animal Instincts is explicitly for traits, which are explained in the Core Rulebook. Simply put, there is no rule that states or implies that traits like Trip grant creature abilities like Knockdown.
Unlike traits, Creature Abilities are explained in the Bestiary and the relevant ones are Grab, Knockdown, and Push. For example, Knockdown requires that:
The monster’s last action was a success with a Strike that lists Knockdown in its damage entry.
But the only things listed in the Table 3-3's damage column are weapon die sizes and damage types. Grab and Push have similar unmet requirements too.
Traits Aren't Redundant
Traits such as Grapple, Shove, and Trip offer four benefits, and the Animal Instinct Barbarian can take advantage of two or three of them. For example, Trip states:
You can use this weapon to Trip with the Athletics skill even if you don't have a free hand.
Many animal attacks do not require a hand at all, for example, the wolf's Jaw can be used to Strike or Trip even when both of the Barbarian's hands are occupied. While Animal Instinct Barbarians cannot use weapons, there are plenty of reasons their hands might be occupied, such as shields, potions, climbing, and much more.
This uses the weapon's reach (if different from your own)
The Tongue and Antler attacks gain Reach at level 7, allowing the relevant maneuvers to be executed with that Reach.
and adds the weapon's item bonus to attack rolls as an item bonus to the Athletics check.
The Animal Instinct Barbarians will eventually have item bonuses to their attacks rolls via Handwraps of Mighty Blows. In particular, item bonuses to attack rolls are usually acquired sooner than item bonuses to skill checks like Athletics.
If you critically fail a check to Trip using the weapon, you can drop the weapon to take the effects of a failure instead of a critical failure.
The animal attacks cannot be dropped to avoid critical failures, so this is the only benefit that Animal Instinct Barbarians can never use.
Feats Like Creature Abilities
Barbarians have several feats that are similar to creature abilities and you can select these feats at the appropriate level, namely Knockback imitates Push and Furious Grab imitates Grab.