I can add Chinese translations to month names. That's easy: cardinal number 🔢 + month 🌙. Lemme try this now.

I was originally going to add them but didn't know if that was meaningful for Korean, Japanese, or Chinese since its just # Month. Is that common writing in Chinese, or is it more common to just do YYYY/MM/DD? If # Month isn't that common, then I see no issue with excluding that.

I'll focus on my native language, and leave the Korean & Japanese part coz I've never learnt them.

As a Chinese l10n, I recommend using Arabic digits YYYY 年 MM 月 DD 日 (e.g 2021 年 7 月 19 日), or without the Chinese characters YYYY/MM/DD or DD/MM/YYYY. Any one of the three should work perfectly in daily situations. Here's the code for the 1st option.

<! -- layouts/_default/date/date.zh-tw.html -->
{{ .Date.Year }} 年 {{ .Date.Day }} 月 {{ .Date.Day }} 日

These three Chinese characters are the same in 🇹🇼 Traditional Chin. and 🇨🇳 Simplified Chin.

[TL;DR]

It's okay to put YYYY/MM/DD or DD/MM/YYYY, except in academic exams, which favor month names like "19th July, 2021" written in Chinese characters YYYY 年 MM 月 DD 日 (with YYYY converted digit by digit, and MM and DD converted as it's numeric value, for example "二零二一年七月十九日")---that's a nice Codingame programming question at beginner's level. I prefer Arabic digits "123" over Chinese characters "一二三" to avoid coding an array of Chinese characters and doing if-else statements. Here's the logic.

1. Writing a mapping/dictionary from Arabic to Chinese characters$chin_num.
2. Then use the $chin_num dictionary to convert YYYY (e.g.2021to 二零二一)
3. month_names[month] 月
4. switch $day
   • case <10: chin_num[day]
   • case 10-19: "十" + chin_num[day%10]
   • case >19: chin_num[day/10] + "十" + chin_num[day%10]

Unluckily, Hugo doesn't provide switch statements. Apart from lacking time to search for the right Hugo syntax for that, maintenance would be a challenge. I found both substr and slicestr for getting substrings. In case that one of them becomes deprecated, and I (or other contributor) would have to refactor the code.

<!-- I suggest using Arabic numerals with 年月日 instead of pure Chinese numerals as follow. -->
{{- $month_names := slice "一" "二" "三" "四" "五" "六" "七" "八" "九" "十" "十一" "十二" -}}
{{- $month := sub .Date.Month 1 -}}
{{ $day:= .Date.Day }}
{{ $year:= .Date.Year }}
{{- $chin_num := slice "零" "一" "二" "三" "四" "五" "六" "七" "八" "九" -}}
<!-- do logic here -->
<!-- render result -->

The three groups of Arabic digits YYYY, MM and DD can be separated by hyphen - or slash /.

From my experience, French month names begin with a small letter. A relevant Hugo page gives an example:
https://gohugo.io/content-management/multilingual/#customize-dates.