See the previous post on the matter:
The flight deck compartment is directly taken from the Left pack only
after passing FL170 why cant we just turn of the left pack?
Because the flight deck compartment would not get ANY air if we were to take the LEFT PACK - OFF.
By ensuring the available (Engine) Bleed supply is feeding the LEFT PACK, we ensure pilots get some air to operate the aircraft. Feeding the Left Pack from the Right Bleed will ensure air is available in the Mix Manifold AND the Flight Deck Compartment.
If we only feed the RIGHT PACK as you are asking, the Mix Manifold will distribute air only into the passenger cabin but
NOT into the Flight Deck Compartment (by design).
Thus to ensure the two people in control of the aircraft get some air as well, in (almost) every scenario we operate the LEFT PACK and turn the RIGH PACK - OFF.