I added it using add_rewrite_endpoint(). According to the Codex, this function also creates the necessary query var(s). Also in other articles, I have not found a statement that endpoints are not supposed to work on static front pages. EP_ALL should include those, as far as I see it. Therefore, I consider it to be more of a bug at the moment.
The solution with creating a custom rewrite rule just for the front page should work, nevertheless.
Paddelboot