Yes. Fact fluency allows higher level of abstraction. This is very clear in the case of mathematics - try doing algebra without having your addition and multiplication tables memorized, kids who fail algebra are often missing these basic facts, along with other basics like order of operations.

It's not just math either. Try to be an electrician without memorizing the basic principles of electricity and how they interact. You're going to have a bad time.

By induction we can see how this scales upwards, basically infinitely, to higher and higher levels of thinking.

I think this is mostly compounding confidence at play that unlocks your mind to be more fluid when faced with a task to do/problem to solve that actually requires skill (not just following a series of known steps). It’s like public speaking (unless you have really bad stage fright) you generally will give a better presentation if you really know your stuff, or at least will be less nervous allowing yourself to get into that flow state.