I actually archive any sort of short-form writing that I think is valuable in my site :) In fact, I compose Twitter threads first as a regular post where each paragraph fits in a tweet, then copy/paste those onto Twitter, and then share the link to the "real" post at the end (to avoid my content being slurped into "thread reader" apps).

But there are lots of short, one-off random-thought tweets that are not worth archiving other than for downloading a copy of your data from these services and saving it offline.