My project today was to book two tickets from Stl to Asia leaving around September 14 plus or minus three days and returning September 23 plus or minus two days.
For out bound I checked all ord-nrt, pek and pvg flights, lax-pvg, and dfw-nrt.
The returns where pvg-ord, nrt-dfw and ord and pek-ord. I used expert flyer for data.
In a perfect world I was looking for either a Business class I fare to upgrade at a reasonable price for me and a F saver for my companion or a coach ticket upgraded to business for me and a business saver award for my companion. The I fares were all well over 6k so I focused on coach tickets and business saver awards.
The results were interesting and consistent with the OPs. There was not a business saver award in either direction on any route on any day. There were plenty of F saver awards but I was no longer interested. What I found fascinating was that even with the lack of business saver awards there was plenty of upgrade space from coach to business. As long as you are willing to fly out out and back in the middle of week- no Fridays, Saturdays or Sundays- booking was not a problem.

I ended up with two coach tickets: Stl-Lax-Pvg outbound and Hkg-Nrt-Ord-Stl all in business except Hkg-Nrt on Cx in coach for $1359 each and 4 evips. I could have saved $60 by flying Jal on the coach route but I prefer Cx. Also, I could have added the Pvg-Hkg leg for another $105 each ticket but I need the flexibility so chose not to.
This worked fine for me but the OP was right about AA not having any Business mile saver availability for the times I checked. Makes me think AA would rather have the coach ticket price and either miles and copay or evip then just trading miles for a business saver award.