PSA The "Have United Call Me" option leads to the web support desk.
I found that they can book a ticket, but don't know everything about booking (such as the free stopover I just mentioned upthread) and one rep was very rude (hung up on me) and unhelpful. On my 2nd HUCA, I just asked to be transferred straight to a booking agent, who gladly booked my free stopover and waived the phone booking fee.