The main problem with your thesis is that Iverson notation was used for at least 5 years prior to being turned into the programming language APL, purely as a mathematical notation.
See also the 1964 IBM Systems Journal document in Volume 3, Number 3, "A Formal Description of System/360" with Iverson, Falkoff and Sussenguth as authors.
I don't see Iverson's intentions as relevant here. He may have wanted APL to be a tool of thought, but it wasn't really picked up as such. APL the computational tool, and its descendants, can claim some successes, but the "tool of thought" concept didn't gain traction for the vast majority of technical people.
It's like if an important human rights document was written in Esperanto. The document might be indisputably important, while its language and the intentions behind it are an incidental historical detail.
See also the 1964 IBM Systems Journal document in Volume 3, Number 3, "A Formal Description of System/360" with Iverson, Falkoff and Sussenguth as authors.