Compilers have parsers, not languages.
Many languages have more than one compiler (e.g., C), and the various compilers may use different techniques for parsing.
Indeed. But consider the 2nd sentence, the conclusion, and the twitter post.
On the other hand, I certain agree with your idea about teach/using handwritten recursive-descent parsers.
Here's an old book that presents it pretty clearly, along with a nice approach for error handling. Maybe you can find it in a library. https://www.springer.com/gp/book/9783540082408