At what rate to the new classes gain additional proficiencies as they advance in level?

I would say the same as the classes from CR P56: Starting Profs & Gaining Prof
Start: Adventuring + 1 class prof + 1 general prof + INT-bonus general profs
Gain: 1 prof at level 5, level 9, level 11
Gain: 1 class prof at each full 2-point save progression (essentially advancing to the last level listed at every other line of the class’s attack and saving throw table)

That’s correct.