As a brief example, suppose the only possible LLM outputs were "hello world", "food", "hello", and "good day" (and that they're all equally probable with no prompting). Suppose your grammar requires a space in the output somewhere and has no other constraints. If you sampled LLM outputs till something passed the grammer you'd receive "hello world" and "good day" with equal probability. If you apply the website's technique you'll receive "hello world" twice as frequently as "good day".
The core problem is that an answer prefix might have been extremely unlikely to yield a valid response, but the technique (probably -- assuming it succeeds -- my example assumed retries would eventually succeed) constructs a valid response from it regardless. Assuming enough independence in the right places everything is fine and dandy still, but correlated errors compound quickly in autoregressive models.
As a brief JSON-specific question, is an LLM more or less likely to make factual errors (hallucinations, truncated strings, missing main characters, ...) when it produces a response failing to adhere to a schema? If factual error rate relates nontrivially to schema error rate then this path is more perilous than it seems. Given the outsized impact certain words or schmooshed together word-phrases seem to have on LLM output, I'd be surprised if details like schema adherence didn't bleed into other characteristics of the output.