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.
You used a calculator every time you needed to multiply or divide simple numbers while reducing matrices to echelon form in introductory linear algebra? Or when doing simple arithmetic necessary to find integrals and derivatives in calculus? I bet you in fact did have those tables memorized, even if it wasn't a conscious effort.
I guarantee you that you've memorized these tables, you just didn't put any effort into deliberately doing it. It happened accidentally as you used numbers frequently.
Of course, maybe I'm wrong. Maybe you DO need to use a calculator to quickly tell me what 8 * 6 is. But I would be highly surprised if you got a degree in math and still needed a calculator for that.
For 8*6? I definitely don't know either of the 8 or 6 tables by memory.
I of course know how to figure it out. Let's see my instinct might be to do 8x2 first because I do know most of the 2s. Then 16x3 I can do in a few steps of addition. 20+12+16 then 48
And yes then I checked on my calculator because I'm posting this online and don't want to look like an idiot.
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.