I think the reason is just that it isn't widely used. It's a very niche use-case, anyway. Even link-shorteners don't particularly benefit from something like this, and for all cases where the actual amount of data is concerned, simple base64 is more efficient.
That is fascinating! In particular, base2048 was designed after twitter adjusted the tweet limit to 280 characters, but started counting “heavy” codepoints as double, making the previously used base65536 less efficient in the context of a tweet.