Perhaps we should restore bookmark--jump-via to its previous behavior and better document the "rules of the road" for bookmark handlers. For simple file- and point-based bookmarks, handlers need to ensure that when they return, the selected window and current buffer are what's intended. For bookmark handlers that perform other actions, those rules need not apply to leverage the bookmark infrastructure.
eww's handler could simply do this itself since it seems eww's url opening behavior warrants this:
(defun eww-bookmark-jump (bookmark)
"Default bookmark handler for EWW buffers."
(save-window-excursion
(eww (bookmark-prop-get bookmark 'location))))
I'd also suggest that we recommend adding an additional property to a bookmark-handler function that could inhibit bookmark--jump-via's call to display-func entirely. In our case, when called programmatically, we use #'ignore, but when the bookmark menu is used, we'd prefer to skip the default pop-to-buffer-same-window.