Actually, I don't think that your assumption is correct. I wholeheartedly recommend you go ask r/linguistics for better answers or check out this excellent video explaining the history of writing, but alas, here was where you asked, and I shall try to give my best answer.

Firstly, there are cultures outside of East Asia that developed symbols for words. The most famous one is the Egyptian Hieroglyphs, though there are/were many more, such as the Sumerian cuneiform, the Maya Glyph, the Aztec writing system, et cetera. On the other hand, not all East Asian culture use individual symbols for whole words, chief among them being Korean (whose system is an alphabet, just rendered into two or three letter long 'blocks'), Manchuria (who uses another type of alphabet, this time written top to bottom, very similar to Mongolian), and to some extent, Japanese (who uses a mix of two syllabaries (Hiragana & Katakana) and the Chinese script/Hanzi.

Technically speaking (and very much simplified), Hanzi, Hieroglyphs, Sumerian cuneiform and Maya are called Logograms. Logograms doesn't necessarily create a symbol for each word; from what I know, these writing systems usually only have a small set of "base" ideas and sounds they deem important, such as a syllable, which they derive their writing systems from. From the video I listed above, an example of this would be when the word "sun" becomes the base from which the words "son" and "soon" through the use of modifiers.

As for the reason for its prevalence in East Asia? I guess its due to the fact that the majority of cultures derive their writing system from another culture. Nearly all languages nowadays, and I do mean the vast, vast majority of languages derive their main writing systems from just two ancestral ones (if I recalled correctly). These are the Egyptian Hieroglyph and the Chinese writing system.

In the case of the hieroglyphs, the letters themselves were simplified over time to make it simpler to write, resulting in glyphs like demotic, which symbols barely look like their hieroglyphic ancestors in the same way that Chinese letters don't really look like images from which they came from. Then, this symbol got co-opted by other cultures, which, due to the difference in language, are now used to represent things that SOUND the same like what it did in the original script, not what they MEANT in that language.

Then, these scripts were, in turn, adapted by other cultures, who did the same thing, plus added or changed the meaning of a few symbols to add sounds that the original script lacked. Rinse and repeat many, many times and you get Phoenician-derived letters such as Latin and Greek, Aramaic derived ones like Arabic, Mongolian and Hebrew, and Brahmic derived ones like Hindi, Balinese and Baybayin.

In the case of Chinese, it was a case of "the language fits the writing system" that made it endure. Chinese is very isolative, so its words don't change form to show a certain grammatical position or feature, such as plurals, gender or so on. Then, most Chinese words are made out of one syllable with a few differences in tone, aspiration, and in some languages, end consonants. Since the word "ma" can mean 'mother' or 'horse', the writing system becomes a good differentiator between words with similar pronounciation.

Since the language is still used by the descendants of its creators, and due to the fact that throughout most of history, China has almost always been the cultural hegemon of the region. So, it didn't really go through a lot of what the Hieroglyph went through, since the transfer of script usually happens with the transfer of the language as well due to its cultural prestige. This either gives the target language a large Chinese influence (Japanese and Vietnamese) or outright making them extinct (Yue). Doesn't mean that it didn't go through the same process at all; Japanese is proof of this (Hanzi was adopted to write sounds as well, then over time it got simplified to Hiragana and Katakana).

Why only those two? Well, it basically boiled down to a game of empire. In the Americas, for example, we have the Mayan and Aztec scripts, which fell out of use because of the Spanish. Same goes with Africa and Insular South East Asia, whose indigenous languages are now mostly written in a derivation of Latin or Arabic. Sumeria was eventually conquered and assimilated by Aramaic (the script family) users as well, ending the era of logographies there.

Hope my answer helps, and feel free to correct/add to my answer in the replies!

Great question, and to answer it, I'm going to start by looking at the origins of our modern writing systems.

It is the common consensus among linguists that there are only four writing systems which were developed independently - that is, with out prior knowledge of writing. These four systems are Sumerian cuneiform, Egyptian hieroglyphs, the oracle bone script, and the Maya script. It is from these four that that the majority of other writing systems have evolved. Namely:

From Sumerian cuneiform came Akkadian cuneiform, and from that came Old Persian cuneiform.

From the oracle bone script came Hanzi, and from that came Kanji and Hanja.

And it was from Egyptian hieroglyphs that systems such as the Greek, Latin, Runic, and Cyrillic alphabets, the Hebrew and Arabic scripts, and Brahmi and its descendants (e.g. Devanagari, the Tibetan alphabet, the Javanese script, all evolved.

Note that these lists are not exhaustive, but the Hieroglyphic phylogenetic tree is by far the largest.

(Unfortunately, I'm unfamiliar with the evolutionary history of the Maya script, so I am having to leave it out.)

Now, let's take a moment and look at the Sumerian line. Sumerian cuneiform, which was used to write the Sumerian language, is a logographic script, meaning that most of it symbols consist of logograms, characters representing words. More specifically, it is a type of logography called a logosyllabary, because it also has syllabograms, characters representing syllables, in addition to its logograms. Although it had access to these phonetic characters, it was largely written with the logograms. When speakers of Akkadian started writing their own language they adapted Sumerian cuneiform into Akkadian cuneiform. This new cuneiform was also a logosyllabary, except that it used its syllabograms much more frequently and had a much smaller inventory of logograms that were used (I've seen it quoted that Akkadian cuneiform texts were on average 60% composed of syllabograms, although I don't recall the time period(s) this statistic applied to, as the use of syllabograms vs logograms did vary over time Correction: according to the Blackwell Encyclopedia of Writing Systems, Akkadian writing tended to be between 86% and 96% syllabograms, compared to Sumerian which tended to be 60% logograms). Its child system, Old Persian cuneiform (used for the Older Persian language), was not a logosyllabary, but instead a syllabary proper, being written almost entirely in syllabic characters and retaining only a handful of logograms (it had about as many logograms as we commonly use today in English writing).

The oracle bone script was the script used to write Old Chinese. Hanzi, its descendent, was also used to write Old Chinese, and is still used for the modern Chinese languages. Hanzi is also a logosyllabary, and similar to Sumerian Cuneiform it is written largely using logograms. Japanese writing is somewhat more complicated, as it actually uses three writing systems mixed together. It uses kanji, the logography evolved from Hanzi, to write most word roots, and hiragana, a syllabary evolved from Kanji, to write prefixes, suffixes, and a couple common words. It also has katakana, another syllabary evolved from kanji, used mostly to write non-Chinese loanwords. Historically, before the creation of hiragana and katakana, kanji syllabograms known as manyogana were used instead, and in fact it was from these syllabograms that the modern kana systems evolved. As we can see, the evolution of the Japanese system somewhat the Akkadian system, with its increased use of syllabic characters.

Finally, let's turn to the evolutionary lineage of hieroglyphics. Unlike the other three (the Maya script also being a logosyllabary), Egyptian hieroglyphs were a logoconsonantary, meaning that it used consonantograms, characters representing single consonants, in addition to its logograms. Descended from Hieroglyphics was a writing system called Proto-Sinaitic, which was used to write several Semitic languages outside of Egypt. Proto-Sinaitic was an abjad, a type of writing system consisting mostly or entirely of characters representing consonants (modern abjads include the Hebrew and Arabic scripts, and indeed both of those are descended from Proto-Sinaitic). And here again we are seeing the same pattern as in the other families. Hieroglyphics used logograms and consonantograms; its child dropped the logograms and just used the consonants. From Proto-Sinaitic sprang several other writing systems, one of interest being the Phoenician abjad.

It was from the Phoenician abjad that the Greek alphabet evolved, an alphabet being a writing system which uses characters representing individual phonemes (vowels and consonants). The Afro-Asiatic languages, which include Ancient Egyptian as well as the Semitic languages (such as Phoenician), have a unique structure for their root words and word morphology, due to which it makes plenty of sense to just write down the consonants in a word, which is why Proto-Sinaitic and Phoenician were abjads and hieroglyphs were a logoconsonantary. The Greek language, however, is not an Afro-Asiatic language, and has more need to write down its vowels, so when the Greeks borrowed Phoenician writing they adapted the abjad into an alphabet. Later on the Greek alphabet begat several others, such as the Latin alphabet (which is what we use today to write English), the Runic alphabets, and the Cyrillic alphabet.

Also descending from the Phoenician abjad is the Brahmi script, used to write Sanskrit. And from Brahmi dozens of scripts used today across India and Southeast Asia evolved.

So, from the above there are 3 key takeaways: firstly, the earliest writing systems were logographic; secondly, as writing systems are adopted for use with new languages, they were also adapted for use, and in particular more logographic systems are adapted to be more phonetic; thirdly, most writing systems in use today evolved from Egyptian hieroglyphs, notable exceptions being Hanzi and Kanji. And based on those three things we can develop answers to your questions.

Why do so many languages use alphabets, abjads, or abugidas?

Because those languages are descended, through several intervening languages, from hieroglyphs.

Why do Chinese and Japanese use logographies?

Because there are not very many intervening languages between them and the oracle bone script.