"Why not just install these as man pages in their own section and read them with man?"
Because downloading the default program is much easier, most users aren't that concerned about adding a small CLI program on their computer, and most aren't that purist about right way to read documentation is through manpages.
I don't think tldr is a small CLI program unless you already have Node.js installed. :)
And I also don't see why installing that is easier than installing man pages, if appropriately packaged. (I'm not claiming they are currently appropriately packaged, I'm just saying that's what they could have done instead of writing a client.)
I just found that and installed it after reading this thread. It looks like on first run, like "tldr tar", it calls "git clone" on some repo. Thereafter, other incantations like "tldr grep" use the already local repo. And the repo can be updated with tldr --update. Seems very fast.
EDIT: My idea is of course not new, see https://github.com/joelekstrom/tldr-man for a way to convert these docs to man pages.