All seats are available on all flights for award travel, admittedly at (sometimes, but not always) extortionate rates.
The new AnySeat program is too new to enable anyone to comment - at this stage - regarding the availability of seats in the (old) Classic award category.
Give it a few months, and we will know more.
FWIW, availability of QF Classic Award seats should match that of AA awards on QF, the only difference being how far ahead you can book.
Sorry if this is not much help.
Dave