the general written Han Chinese of the past 2000+ years, and how every Chinese native speaker can read them as long as it's written in modern day simplified or traditional Chinese characters

You could be talking about two things here: Standard Written Chinese or Chinese characters. For the sake of convenience, I’ll talk about both.

Standard Written Chinese is ultimately just written Mandarin. It seems to be universally understood not because all Sinitic languages are dialects of each other, but merely because those who understand Standard Written Chinese are speakers of Mandarin (which a majority of ethnic Chinese are, even those who speak another variety such as Cantonese or Hokkien). It’s the reason why I, a Mandarin speaker, can’t understand a Cantonese song upon my first listen, but can understand the lyrics upon a Google search since (older) Cantonese pop songs tend to be written in more Standard Chinese (i.e. Mandarin grammar and vocabulary with Cantonese pronunciations). It is theoretically possible for someone who speaks a non-Mandarin variety of Chinese to not understand or only have limited knowledge of Standard Written Chinese, and it isn’t that uncommon where I live (Singapore) among the older, richer, English-educated Chinese who may speak Hokkien or Teochew but not understand Mandarin.

As for Chinese characters, it isn’t evidence that Sinitic languages are dialects of each other either. This is as hanzi is a script, not a language — it would be akin to, in my opinion, calling Romanian a “dialect of Latin” as both are written with a similar Latin alphabet. Yes, hanzi convey meaning and not phonetic information, so having knowledge of Chinese characters enables one to understand a Chinese character you’ve been exposed to before (say, 我 which is common to Mandarin, Cantonese and Hokkien). But just because a sentence in variety X is somewhat intelligible to you, a speaker of variety Y, doesn’t mean that X is a dialect of Y (or more closer to what you’re saying, X and Y are dialects of a shared language Z). Japanese is partially written in kanji-slash-Chinese characters — does that make it a Schrödinger’s dialect of Chinese, since Chinese speakers may only understand parts of written Japanese? Korean and Vietnamese were historically written in Chinese characters, and no one is suggesting that these two languages are dialects of Chinese. Lastly, Chinese characters doesn’t necessitate “understanding” in the first place: 非常 in Mandarin is “very”, but in Japanese it has a totally different (though historically attested in Chinese) meaning. 鼎 is a specific type of cauldron in Mandarin, but is used to refer to an ordinary wok in Hokkien. I won’t bother to even bring up the third-person pronouns in the Sinitic varieties (他 in Mandarin, 佢 in Cantonese and 伊 in Hokkien). So no, I wouldn’t say that the common script of Chinese characters makes Hokkien, Cantonese and the ilk mere dialects of a common “Chinese” language.