You do need to set eldoc-documentation-strategy to
eldoc-documentation-compose though (this should really
be the default).
Huh, I wonder why I can see both flymake + eglot in the eldoc doc
buffer when my eldoc-documentation-strategy is the default value?
Because Eglot changes eldoc-documentation-strategy automatically. It
shouldn't but the default value is really bad.
The reason the default value is historic. Previously, there was a
single producer of ElDoc, and only in Emacs Lisp. It would decide
whether to show variable _or_ function doc, even if a given symbol had
more than one meaning. So what's the problem with setting
eldoc-documentation-strategy something like e-d-compose, you ask.
Well, because of the default value of eldoc-echo-area-use-multiline-p,
people would be seeing "bouncing" in the echo area while editing Elisp,
which is something they are not used to.
I think a very good solution is to set e-d-strategy to e-d-compose
globally and e-e-a-use-multiline-p to 1 in emacs-lisp-mode.
I once proposed this in this bug tracker, but the message was garbled by
some side discussion, and I gave up. And ElDoc wasn't so powerful then.