(By the way, I am in considerable agreement with Igor Rivin's premises and observations in that essay, and many of the in-principle conclusions, but might tend to doubt that the course of the juggernaut that is American culture and the concommitant educational system can be altered by the actions of a few quasi-rogue mathematicians in universities...)
So, first, my comment/reaction is that it's maybe not that a program is a proof, but the demonstration of correctness of an algorithm certainly is.
For context in regard to that article: I, too, have been frustrated by the intellectual state of undergrads, from perceptions of calculus, and I. Rivin's example of linear algebra, and even worse with "intro to abstract algebra". The latter was supposedly aimed at future high school math teachers, future actuaries, and math majors not going to grad school in math. After several experiments, it became clear that a majority of students could not only not write proofs or logical arguments or even sensible arguments, but could not reliably compose coherent paragraphs about very mundane actions.
Motivated by this situation (and other things), I cobbled together a "crypto" course and an "error-correcting codes" course to allow discussion of abstract algebra (and other important, basic mathematics along the way: combinatorical probability, a little linear algebra, finite fields, ...) with practical applications, and algorithm-oriented.
I still did ask students to narrate their execution of algorithms (with human-scale data). I had thought this would be a fair-if-minor challenge/goal.
I was to a considerable degree amazed that even very many "good" students could not, even with prompting, say what they had done, or why, even in very simple computations. "It's obvious." I tried to get them to write as though they were explaining something over the phone to someone who had no idea, but this seemed ... impossible.
That is, as artifact of the educational system in the U.S., people cannot reliably explain a process or algorithm. Observation suggests that a decade or so of not being required to explain things, but only to follows rules, does not beget narrative skill.
Nevertheless, I strongly hesitate to describe what even mathematicians would like from students as "proof". "Explanation", as a less stylized, more colloquial label, might be less polarizing and more explanatory to the general population.
But should kids be required to write programs? Maybe in pseudo-code. While I do understand Scheme and other dialects of Lisp and some other languages, I'd hesitate to insist on the "purity" of Scheme, as opposed to Python, for example. Not to mention the opportunities for killing time wrangling with syntax that is wildly different from English. No real point in that, unless we are wanting to inadvertently return to the (popular!) idea of mathematics as a discipline of adherence to semi-inexplicable "rules".
I was also dismayed to learn that many, many students believe that computers can do things that are even in principle impossible for humans, not merely on grounds of time or memory... but that they (computers) can compute trig functions and detect primes and do all sorts of things that no human could conceivably do. There is scant understanding that the atomic steps are perhaps even smaller for computers than for human-scale operations. Computers are magic?
So, yes, it would be good for U.S. students to learn that "proof" might mean, for example, explaining why an algorithm (whether executed by them or by a computer) produces useful outputs.
And then there're probabilistic algorithms (very relevant in crypto and in coding, thus to the internet, thus to everyday life...), which reliably disturbed many of the math majors... :)