No they're right. The innate immune system plays an important role in the start of an infection. When it is busy trying to kill one thing you are at slightly higher risk of a different infection- this is largely why secondary infections are a thing.

What you are speaking about is the adaptive immune system which kicks in a bit later, and is certainly more powerful, but a functional innate immune system should prevent a lot of infections before the adaptive response has really had to worry about it.