What is your algorithm for determining which sentences are meaningless? Since we don't have such an algorithm (without serious flaws), I'm guessing your algorithm is probably flawed also,

The "beliefs should pay rent" heuristic mentioned by User:Tiiba already answers this. My method (not strictly an algorithm[1], but sufficient to avoid paperclip-pumps) is to identify what constraint such an expression places on my expectations. This method [2] has been thoroughly discussed on this internet and is already invoked here as the de facto standard for what is and is not "meaningless", though such a characterisation might go by different names ("fake explanation", "maximum entropy probability distribution", "not a belief", "just belief as attire", "empty symbol", etc.).

Is your claim, then, that the "beliefs should pay rent" heuristic has serious enough flaws that it leaves an agent such as a human vulnerable to money-pumping? Typically, beliefs with such a failure mode immediately suggest an exploitable outcome, even in the absence of detailed knowledge of the belief holder's epistemology and decision theory, yet that is not the case here.

With that in mind, the excerpt you posted does not pose significant challenges. Observe:

it is not an adequate solution if it answers the question, “Why is the Liar Sentence meaningless?” simply with the ad hoc remark, “Otherwise we get a paradox.”

This was not the justification that I or User:Tiiba gave.

For example, the sentence, “This sentence is in English,” is very similar to the Liar Sentence. Is it meaningless, too?

The claim that a symbol string "is in English" suggests observable expectations of that symbol string -- for example, whether native speakers can read it, if most of its words are found in an English dictionary, etc. This is a crucial difference from the Liar Sentence.

What ingredients of the Liar Sentence make it meaningless such that other sentences with those same ingredients will also be meaningless?

Again, lack of a mapping to a probability distribution that diverges from maximum entropy.

Are disjunctions with the Liar Sentence meaningless?

The non-Liar Sentence part of them is not.

The questions continue, and an adequate solution should address them systematically.

The requirement that beliefs imply anticipations is systematic, and prevents such a continuation.

[1] and your insistence on an algorithm rather than mere heuristic is too strict here

[2] which is also an integral part of the Clippy Language Interface Protocol (CLIP)