So, I tried to think of counter-examples to "most (and possibly all?) early application of mathematics required computation" just for the sake of discussion.

I think I have only one, which is the establishment of axioms, both philosophically (as a method) and specifically (e.g. in Elements).

A revisionist history might say that choosing axioms doesn't require any computation, just a keen sense of style and close observation of the world.

But actually, I'm sure that the choice of axioms was a long and drawn-out process informed mostly by computation and checks that the computed values/proven theorems matched with physical intuition. After all, that's kind-of how it's done today, even by people who have lots of experience with formal systems.

Now I really want to read pre-Euclidean mathematical philosophy to see if I'm correct :-)