The doc string for select-window does not discuss the sometime need to also set input focus to the selected window's frame when one wants future input events to go there.  I would suggest adding this along with a reference to select-frame-set-input-focus.

Even better might be to add select-frame-window which would combine select-window and select-frame-set-input-focus for those times where you want to both select the frame and change its selected window.