I wouldn't consider someone who's new to Excel and doesn't use formulas to be a programmer. But having seen some truely bohemoth spreadsheets from finance (and being asked to port them), I'm having a hard time drawing a line between the two.
To share your analogy: many folks have trouble with algebra. It's not that it's categorically impossible for them to do it; there could be any number of factors (bad math education, home troubles making school more difficult, etc.).
With a lot of time and dedication many of these folks could eventually get pretty good at doing algebra. I've tutored these folks before. Some of them have enough time and dedication to get better at it. But without that (and at least time is outside most of these folks' control), many of them will struggle.
So I don't think the parent comment was saying "Most people can't program in the same way a gorilla can't speak English", but instead "Most people can't program because they don't have the acquired skills to program." I would add that these folks' goal is often not programming but their original job.
After all, language is for the precise and concise communication, human to human or human to machine.
In my experience, it isn't. Computers are much more pedantic and less forgiving than other people. Computers won't understand "what you meant to say".
Additionally people often have trouble communicating effectively with other people in their natural language. So a DSL being in that language doesn't necessarily guarantee the user's success in using that DSL correctly.
I see programming mostly as a mindset. Like doing algebra. You couldn't do it as an infant, because you just didn't understand the concept of using letters instead of variables.
Of course, there usually is a level of complexity that you need to get used to as well, but you can also see this as a defect of most API's, or as an inherent quality of doing anything nontrivial.