I don't think that's true for any chinese languages other than mandarin and cantonese. I've been in Taiwan for the past few weeks and dipped my toes into Hokkien/Taiwanese/Souther Min, and the situation there is fascinating. In every day normal situations it's never written, if two native hokkien speakers want to send an email or text it will be done in mandarin. In the rare occasion Hokkien is written, it will be done haphazardly. Sometimes you'll choose a mandarin character that kind of means the same thing as a hokkien morpheme, and sometimes they'll choose the mandarin character that kind of sounds the same the same as a hokkien morpheme. Native speakers will be able to read this, but often struggle.
I have read that people used to be able to write Hokkien in characters, but that's lost now. I'm confidently told by native speakers today that the language has no written component (沒有文字). There's romanisation system that seems to be able to represent every sound prefectly, and more has actually been written in romanised hokkien than by any other methods, but Taiwanese people have a bit of a phobia of the latin alphabet, and no one uses it.
But I'm actually unclear about this.
The weird thing is that certain dictionaries give pronunciations of hanzi in multiple Chinese languages (e.g. Wiktionary) including "scriptless" languages.
I've also seen cases in Taiwan where e.g. a restaurant name or place name is written in hanzi and English and the romanization is clearly not using the standard Mandarin pronunciation of the term. An example of this is 淡水 which is romanized "tamsui" in Taiwan but would be "danshui" in pinyin or "tanshui" in Wade-Giles. Syllables ending with "m" are not a thing in Mandarin as far as I know, and 水 is never romanized as "sui" (though I could chalk the 水 one up to an alternate local romanization system like POJ, which I don't know anything about).
Finally, I've also seen text in Taiwan with four-bopomofo ruby. From my understanding of bopomofo; for Mandarin you won't have more than three glyphs plus a tone mark -- the initial consonant, optional medial i/u, and optional final. This also gives me the impression that Min Nan can be written in hanzi. Unless Taiwanese Mandarin differs from Mandarin significantly in pronunciation (which may be the case, I don't know).
All this leads me to believe that other Chinese languages can use hanzi, but do so less often, perhaps? This is something I've always been confused about but never really looked into it.
I have a Taiwanese book for school children that includes the governments romanisation, hanzi, and some kind of extended bopomofo. While the native speaker I was learning with could read hokkien-with-characters, he couldn't do it quickly, and often had to puzzle over a certain character. In cases where they had both characters and romanisation I was able to help him on the characters he was stuck on by (badly) pronouncing the romanisation. Still, I haven't met any Taiwanese hokkien speakers at all interested in romanisation. Which is a shame - it covers all the sounds, and the language does seem to be slowly dying (the fact people don't write it probably has a lot to do with it).
Though it seems like hokkien-with-characters is a thing (from what you say, and also from what I've seen with 4-bopomofo ruby and Wiktionary), just less prevalent. Or perhaps one can look at hanzi at the same level as romanization for Hokkien (except romanization is more prevalent), i.e. "someone else's script that kinda works for us so we use it sometimes"
I'm surprised the extended bopomofo (I assume it's https://en.m.wikipedia.org/wiki/Taiwanese_Phonetic_Symbols which actually till now was the thing I thought was bopomofo) is used more as a writing system for Taiwanese.
And that's the thing, they use characters when they have to write it, but even then native speakers struggle to read it because there's no commonly used standard way to do it. In practice, they just write in mandarin.
I call them dialects for lack of a better word. For example, an Algerian Arabic speaker would not be understood by an Egyptian, and a Kuwaiti Arabic speaker would not be understood by a Syrian, etc. Regional dialects may have an accent, but more importantly use different words and may even have different, informal grammatical rules.
When it comes to writing, everyone uses Arabic letters, but each region may write different words depending on the regional dialect.
The entry on Tunisian Arabic[1], which is my native dialect, may provide an idea on how it differs from other dialects.
Read the first two paragraphs for a better idea of what I'm talking about: https://en.wikipedia.org/wiki/Varieties_of_Arabic
But yeah, it's similar to the Arabic situation, but the scale may be different.
"chinese" isn't one language. It's a set of many languages which share some grammar and a common logographic script (though the script isn't used in exactly the same way for each, e.g. Beijing will write/say 哪儿 instead of 哪里 for "where?" -- there are many other differences but I don't know enough to give a clear idea of its scale).
The spoken languages are quite different otherwise, at least as different as European romance languages. There is a "Standard Mandarin (普通话)" however; which the government has pushed and is spoken by most residents of Mainland China.
I'm skeptical that this is a good way preserving the culture. If language does have an effect of preserving the culture; this move would also have effect of killing subcultures from regions (which may indeed have happened in China; though I don't know). China and India are both pretty diverse culturally.
You're making the argument that the popularity of English is detrimental to the preservation of Indian Culture and in the same breath suggesting forcing Hindi on everyone as a solution, as if it wouldn't have exactly the same effect on a culturally diverse country.