Just to answer myself re: above…I finally found a way to get the “old” multi-city booking page.
I have to log in to my Cathay account on either my iPhone or iPad. Then, go to “Book” and select “Multi-city/stopover”. This will then redirect to an external browser (in my case, to Safari), and the multi-city booking page shows up where, for each flight, I can select the date AND THE DEPARTURE TIME (e.g. 06:00 - 11:59 or 12:00 - 17:59, etc.). After all the desired flights are selected, the booking engine will then give fare options for Econ Light, Econ Essential, and Econ Flex…each with its own fare restrictions.