It seems to not be possible to add another user as a watcher via the API.
I can understand why the web UI doesn't allow this: it is a somewhat rare use case (but do note that Jira itself DOES allow this feature, and it is sometimes quite handy).

But for the Rest API, which is by definition used to automate tasks and build tooling, I think it does make a lot of sense to be able to manipulate the watcher list.

My use case is a internal tool that help us with code reviews where you can add rules , e.g. "automatically add Bob as a watcher to any Pull Request that contains Security related code".

So, the same way the REST API allows to add/remove arbitrary reviewers, it would be good to have that capability for pull requests watchers as well.