I wish!
BA Staff do not earn Avios or Exec Club status for staff concessionary travel at all.
They can be members of the Executive club if they pay commercial fares as anyone else and many are Silver and Gold members.
Personally I am a member and often find that commercial fares are cheaper than the staff travel rates, especially if you consider sale fares and book in advance. Remember most of the staff tickets available to us are on a standby basis, so no seats, no travel. Sadly the taxable amount is the same as well, so a standby ticket is no longer the bargain it used to be.
I am also a Tesco and Amex collector....... in fact a bit of an addict!