I think if you regard logic (in philosophy) and maths (as a huge broad field) and computing (specifically a sub-field in maths to some people) its pretty clear that logic and computing have a huge relationship.
I can think of lots of other subfields in maths, which have huge inter-relationships. Applied maths, whats that got to do with probability? Well.. it turns out that modelling complex systems uses Monte-Carlo methods .. (a fictional example, I suspect, I know the manhattan project people dreamed MC up but its modern applicability is unknown to me)
You don't think maths informs programming, or its over-stated? I guess thats true, in as much as poetry doesn't inform legal writing. But, I observe that people who do enough poetry or writing to understand the difference between a simile and a metaphor and an allegory, are really on-point communicators, and the law needs that concision and precision.
I think people with good groundings in maths (and logic) make awesome programmers but its not strictly neccessary to be a mathematician to know how to "speak" in a programming language. What pitfalls you avoid from your knowledge, I cannot say. But I do know that huge pitfalls lie in naieve programming: large loops iterating over un-initialized data structures, not understanding the if-then-else logic or side effects of expressions, tail recursion..
I think computing is a sub-field in maths. How much it matters depends on how much your code matters.