When you are trying to book with A3 you will have to check availability with A3. Whatever availability MileagePlus shows will be different than what other FFP's will show/have access to. This is not only for A3 but applies to all FFPs.
Reason being that every airline that has award seats available can (and will) filter which partner FFPs will get access to those seats. In your example SQ has decided that United is allowed to book those, and Aegean is not.