I don’t think you have to hold that all self-referential sentences are meaningless in order to show that This sentence is false is meaningless. Sentences which refer to themselves qua linguistic entities (strings of words) seem to be fine; e.g., This sentence is in the present tense is probably meaningful (and true). But problems arise when a sentence refers to its own content. In order to understand what such a sentence even means, you need to already know what it means; therefore, it’s impossible to understand it. I don’t think this is “disturbingly ad hoc.”
JohnD: To say that an English sentence is false is to say that its content is false. So This sentence is false really means The content of this sentence is false. But this sentence doesn’t even have an intelligible content, so there’s nothing of which anything is affirmed.