Does the “Rapier Training” Swashbuckler class feature count as having the Improved Critical feat?

I’m DMing for a group of five people. One player, Swashbuckler with the Inspired Blade archetype, wanted to select critical feats which have Improved Critical as a prerequisite. The following Inspired Blade feature describes it as follows:

Rapier Training (Ex): At 5th level, an inspired blade gains a +1 bonus on attack rolls and a +2 bonus on damage rolls with rapiers. While wielding a rapier, she gains the benefit of the Improved Critical feat. These attack and damage bonuses increase by 1 for every 4 levels beyond 5th (to a maximum of +4 on attack rolls and +5 on damage rolls at 17th level). This ability replaces swashbuckler weapon training.

It seems to me that the player cannot do it, but I’ve house ruled otherwise. The reason is because his character looks pretty bad in the game (I did not tell him this) and is already getting behind others on the table. Charmed Life with Opportune Parry and Riposte are letting his immediate actions overloaded whilst his char is too squishy due to spending ability points on Dex, Int and Cha. Taxing him another feat with improved critical seemed bad from a balancing perspective. There is a vanilla warrior on the group who does everything the swashbuckler does but significantly better.

That being said I’ve got two questions for you:

  1. Does the "Rapier Training" Swashbuckler class feature count as having the Improved Critical feat? (From the title)

  2. What would you have done in such situation?

Obs.: this class seems quite unbalanced and badly designed to me.

Thanks!