> Are you starting Emacs as daemon?
Yep.
Trying to follow your suggestion I've written
(defun hide-minibuffer-scrollbar (frame)
(with-selected-frame frame
(set-window-scroll-bars (minibuffer-window) 0 nil)))
(if (daemonp)
(add-hook 'after-make-frame-functions #'hide-minibuffer-scrollbar) ; Only for client sessions
(set-window-scroll-bars (minibuffer-window) 0 nil))
Now client sessions start without the minibuffer scrollbar, but as soon as I use the minibuffer it comes back and it isn't removed afterwards.