Seems that AA has released very few seats to be booked as award flights. However, the root problem with LAN.con seems to be that it shows that there are seats on a flight which doesn't have available award seats. I checked the same dates through avios and the flights I checked were unavailable at avios and at available at lan.com (at later stage the system told that the seats are unavailable for award bookings). One explanation is that lan.com at first checks whether there is available seats in that booking class and at later whether award seat could be booked.

LAN has normally much more available seats with European partners than with AA. I have been able to book awards with BA&Finnair 24 hours before departure on routes with heavy business traffic.