I've found that the Expert Flyer tool has an accurate award search for ET. My personal experience is I put together an itinerary by searching ET award availability segment by segment, verifying that there are 2 seats open on each segment on Expert Flyer. Then I look for the entire itinerary on the UA award search site, it can be booked and ticketed. My success rate has been at or close to 100%. I have also found that as soon as I book it, the ET award availability on those segments disappears on Expert Flyer.

In all honesty, my trips are a bit less complex than the example you are posting. Mine have been all ET flights, on ET metal. Typically ORD-ADD-XXX where XXX is my desired destination or the reverse (XXX-ADD-ORD). If I cannot find availability with ORD as the ET gateway city, I will look at YYZ, EWR, JFK or IAD as the ET US gateway city and hope I find a routing for ORD to/from the gateway city for a full award flight from/to ORD. If not I look at a separate positioning flight and hope I can stitch all it together as a single itinerary, which can be a challenge with the UA call center.