The people created a language with a need for reason (communication), but with no real need (reason) to count. Apparently a person either had and did whatever, or had not and did nothing.

No real need to count people, since everyone was likely known. You either went hunting with Roger, Joe and Sam, or not.

No real need to count mangos, since you either brought home enough food, or too much food, or not enough food.

Recipes? You either cook it, or don’t, or burn it, or cook it a bit longer.

The limited numbering system mentioned in the article might even have been used in a local gaming system for the high-brows of the society, like a local chess club, or even a touch of wizardry. No need to let everyone know exactly how to figure out how many more fish you need to catch for the big party. Becoming the go-to guy for the tough answers is sorta mystical.

Number As A Cognitive Technology: Evidence From Piraha Language and Cognition – Michael C. Frank, et. al.

I guess all those great minds are now finding employment as psychologists and economists.

https://boingboing.net/2020/06/26/a-scheme-of-heaven-is-a-deep-i.html

But eventually we learned that we can’t be trusted to rely on “common sense” because it doesn’t always work. In fact, it rarely works as well as we expect it to.

There’s a story that the Hottentot language doesn’t have a word for any number greater than 3. They count “1, 2, 3, many”. To be fair, in their world, this was all they really needed.

But there comes a point when you really need to believe that 5+7 has to be the same as 7+5 and that 7×5 has to be the same as 5×7, but that 2x(3+4) is different than (2×3)+4. And to convince yourself of that, you need logic.

Also, I never phrased the question as whether counting came before or after arithmetic. The question was reasoning or counting, alternatively phrased as logic or arithmetic.

Dunno what others here think, or what your answer specifically was, but I’mma gonna place you in the “counting” camp.

Seriously, Counting came before Arithmetic. Arithmetic is a set of formal rules and procedures that systematize and extend Counting. For example, even the most primitive tribal culture needs to count things, even add and subtract. More advanced Arithmetic, like multiplication, division and fractions, comes later. 5+7 = 7+5 = 12.
From there on you progress to:

Algebra: the application of arithmetic to abstract, general numbers (as opposed to actual numbers). a+b = b+a = c. In Algebra you don’t deal with numbers, you deal with symbols which can be ANY number.

Real Analysis: (Which includes calculus and differential equations) which studies the behavior and changes of algebraic statements.

Complex analysis: Which introduces the tool of imaginary numbers to help make sense of real analysis.

And of course, there are things like Geometry, the study of space itself and of relationships within it, which doesn’t seem to easily fit in with the earlier progression. Logic plays a big role in Geometry, so my guess is it came first, but I don’t really know for sure.

Its only fair to warn you, I didn’t learn or read this anywhere, its only my opinion. I have a book called “An Introduction to the History of Mathematics” on my bookshelf, but I must confess I’ve never read it.