The linked article proposes resolving the liar's paradox by forbidding self-reference in some circumstances, or questioning the meaningfulness of sentences that refer to "this sentence".

This is somewhat akin to what mathematicians tried to do in response to paradoxes like Russell's, but Gödel was still able to construct self-reference indirectly!

https://en.wikipedia.org/wiki/Type_theory#History

One concern with trying to do this is that it seems to rule out of bounds quite a lot of natural uses of language that talk about the use of language. The original liar paradox referred to a class of persons (those from Crete) as liars, rather than a statement as a falsehood.

https://en.wikipedia.org/wiki/Epimenides_paradox#Origin_of_t...

We could also imagine cases like

"It is impossible to express truth in the English language."

or

"No sentence that contains eight words is true."

or

"No generalization about propositions is correct."

It seems increasingly difficult to maintain that all of these utterances are meaningless, at least on the specific grounds that the article gives.

Or, to borrow a favorite line from my friend Erik:

"Il ne faut pas croire que la parole serve jamais aux communications véritables entre les êtres." ("One should not believe that words suffice for true communication between beings.") -- Maurice Maeterlinck

... which also provides a pretty nice liar paradox if you believe that Maeterlinck was a being, or that I'm one. :-)