It can, but the information is not very helpful to you (rather like the IB examples you posted above):
It shows you some (AA Saver) availability on a flight that could form part of a valid routing, but cannot tell you whether there is anything available on the BA flight.
For travel between the US and Europe (or within Europe) using AA miles, EF wouldn't be my first choice of tool. AA's own website is a better place to start.
Correct, we do not choose the routing, the reservation systems do, and they don't understand what is an award class vs a paid class. The best way to search is to only search the city pair that AA operates when searching for AA award inventory, same for the BA flights for BA award inventory.