The case GP is talking about is (say) having subtitles in Chinese characters (in Mandarin), which can be read by people who speak Hokkien or Cantonese; or having street signs in Hong Kong written in "Written Cantonese", which can be read by people from Beijing.
The answer to the question, "How does someone from Beijing understand someone speaking Cantonese?" is generally, "They don't."
I think GP‘s point is that right now they can read the same text even if they can’t understand it spoken. Moving to a phonetic script would mean they can no longer read the same text, ergo worse
A phonetic/phonemic script would be worse for mutual intelligibility for those privileged enough to know written Chinese, which is notoriously difficult to learn.
But it might be better for literacy in general because it would be much easier to learn. And, though it is a comparably much more trivial issue, better for creating something like a morse code.
That said, I know there are many other issues apart from simple ease of learning that figure in to what language people learn or want to use. Written Chinese has a rich history and culture, and it's tied to people's identity so of course jettisoning it for something arguably more practical with the added downside of loss of mutual intelligibility among those who know written Chinese would likely result in strong opposition.
If you see the counter-arguments are coming from historical and cultural stuff, then you are making wrong assumption here. At least not in this branch. Literally you are the first one to mention this argument.
The answer to the question, "How does someone from Beijing understand someone speaking Cantonese?" is generally, "They don't."