I used to have the same problem as yours so I decided to write a function:
;;;###autoload
(defun my-dired-do-rename (&optional arg)
"Rename/move the file at point with itself as the initial input.
Doesn't include other regular file names while `completing-read'."
(interactive "P")
(let* ((old-file (directory-file-name (dired-file-name-at-point)))
(new-file
(read-directory-name "Rename or move to: "
nil nil nil
(file-name-nondirectory old-file))))
(dired-rename-file old-file new-file nil)))
(define-key dired-mode-map (kbd "r") #'my-dired-do-rename)
Alternatively, you can wrap dired-do-rename around an advice that disables vertico-mode, but that will give up the ability to easily choose another directory to move to.