You're exaggerating the distinction. Pronouncing ся as sa or s-ya does not cause listeners any problems
But it's not about "sa"/"sya", it's about soft and hard consonants in general, and they exist in every single word.
I really don't understand the insistence that all aspects of pronunciation must come right at the start. [...] First the basics, then fill in the gaps, and then refinement.
I think it's the crux of the matter. Most Russians consider proper (or close to it) pronunciation a basic, not an advanced skill. That's simply how we are taught. When we started learning English at school as first graders, one of the very first things we had to learn was IPA, so we could understand proper transcriptions. Not once Russian alphabet was used to explain English pronunciation.