When using the UA search engine, you need to look for Saver business award space, which is labelled as "Business (I)". Generally, those are the only seats that can be booked across all *A partner programs.

IMO the OZ search engine is definitely frustrating, given that you can only search one day at a time. I always use the UA search to find availability, then input the exact route/date in the OZ search, and either book it online myself or call the reservations phone line. I recently booked SIN-TPE-SFO on BR and SFO-NRT-SIN on UA/NH after firstly finding the relevant Business (I) seats via the UA search.

I don't have any experience booking EY awards via OZ, so I'll let someone else help with that.