That's true, but it seems these are practical limitations, not theoretical ones.
The nice thing about the terminating case being a computer, though, is that it's a moving target (Moore's law, etc.). At the extreme case, a compiler could model every molecule in a human brain and use the simulated intelligence to convert the description into a language that it can compile.
I would be crazy to expect that any time soon, but I think it shows that in theory we can do better than existing programming languages do.
By the way, I re-worded the last sentence to "as long as programming languages are less expressive than written English, I'll keep welcoming them.", because I don't think it's inevitable that programming languages can become as expressive as natural languages. I do think that as long as they aren't, we can do better.
In my opinion expressiveness doesn't exist objectively.
With that motivation for language design (achieving the expressiveness of English), I'm afraid you're running to a self-centered opinion; it is expressive only for those who have learned it since childs. The hierarchy of programming languages derives from the fact that expanding those languages is a lot of work (reimplementing the parser), in comparison with a natural one (metaphors, or neologisms). That's why, I think, a more general language (LISP, Smalltalk) is better. They don't give you expressiveness, which is only given by semantics, but at least they don't narrow those semantics. In comparison, English is a very ambiguous language which seems more expressive because it's semantics are huge. Which leads to the problem that, even if you built an artificial brain, you'd have to teach that brain ontology. Ultimately, the problem's root is not found within programming languages, but is an inertia to extend those which exist.
This ignores the fact that even if you explain quicksort to a random person and give them a list of a few thousand numbers they can only look at one at a time you will invariably find out that most of the people will get it wrong most of the time. Just because you think you can imagine the execution of an algorithm given its description doesn't mean any random human brain can; doesn't mean even you can, if pressed, executed a really complicated program without errors.
The error rates we get on modern computers are scary for most "algorithmic" problems.
The nice thing about the terminating case being a computer, though, is that it's a moving target (Moore's law, etc.). At the extreme case, a compiler could model every molecule in a human brain and use the simulated intelligence to convert the description into a language that it can compile.
I would be crazy to expect that any time soon, but I think it shows that in theory we can do better than existing programming languages do.
By the way, I re-worded the last sentence to "as long as programming languages are less expressive than written English, I'll keep welcoming them.", because I don't think it's inevitable that programming languages can become as expressive as natural languages. I do think that as long as they aren't, we can do better.