Grammars are not AI, it's just another formalism (like regular expressions, Turing machines etc.) - formalism alone doesn't solve anything.
In formal language theory, you have different classes of grammars, the most general ones correspond to Turing machines, i.e. they are a glofified assembler and you can do anything. The most restricted (in the Chomsky hierarchy), "Type 3" grammars, are basically another notation for regular expressions, and they described regular grammars.
There are algorithms for learning grammars, but the issue with that is that the induced grammars may not resemble anything that a human may write (in the same way that a clustering algorithm often does not give you the clusters you want).
But to answer your question, we need to separate the discussion between appropriate representation and method to solve a problem.
I believe grammar-based compression - if you accept probabilistic grammars - is similar to LLM-based compression at some level in the sense that highly probable sequence of words get learned (whether by dictionary, grammar, neural network = LLM, could be just an implementation detail). Whichever you choose, you still need to solve the problem you are trying to solve (any grammar formalism still needs a parsing algorithm, and an actual grammar that does something useful - even after you develop a parser generator).
[Side rant, not responding specifically to the parent or OP: as a linguist, I'd also warn everybody to use "AI" with an article: *"an AI" (asterisk marks wrong use). It wrongly suggests human-like properties when it's actually just a matrix of numbers that encode a model. Here is a test whether you are using "AI" right: replace it by "Applied Statistics" in a sentence and see if you would still say it.]
AI is just an academic field (ill-named for historic reasons), subpart of computer science, and while it's fair to talk about useful representations for modeling human-like behaviors, we should focus on what intelligence is, and talk about the limits of concrete models and possibilities to extend them.
The thing about LLMs is they are a bit like the perfect snake oil salesman: extremely articulate, but knows very little nothing about a lot, understands nothing. (Whatever one criticises, they do the one thing that they are designed for very well: to generate text. Sadly that misleads a lot of people that they are just next-word/next-sentence predictors.)