I understand what a CFG is and why Dyck's language (matching parens) is not a regular language. My point was that CFG/CFL is less motivated by a reasonable and uniquely characterising constraint - such as making memory usage independent of the size of an input string - than regex is.

Then again, you are right that CFGs are very natural. And they do admit a few easy O(n^3) parsing algorithms, like Earley and CYK.

I think your last sentence relates to Visible Pushdown Grammars. See also Operator Precedence Grammars.