Can an 8th level Monk/Barbarian (Path of the beast) really make 5 attacks per turn?

In Tasha‘s Cauldron of Everything, WotC released a new Primal Path for the babarian, the "Path of the Beast" which has the following rage feature:

Until the rage ends, you manifest a natural weapon. It counts as a simple melee weapon for you, and you add your Strength modifier to the attack and damage rolls when you attack with it, as normal. You choose the weapon’s form each time you rage:[…]

Claws. Each of your hands transforms into a claw, which you can use as a weapon if it’s empty. It deals 1d6 slashing damage on a hit. Once on each of your turns when you attack with a claw using the Attack action, you can make one additional claw attack as part of the same action.

Combined with 5 monk levels to get Extra attack and flurry of blows, could such a monk barbarian make 5 attacks in one round like:

1st round:

  • rage (bonus action)
  • 2 attacks (action) with claws
  • 1 extra claw attack (from "claws" feature)

2nd round:

  • 2 attacks (action) with claws
  • 1 extra claw attack (from "claws" feature)
  • 2 more attacks (bonus action flurry of blows)

Or am I missing a restriction that prevents this?