Does the supplied door bonding strap mean the complete cabinet installation is already earthed?

No.

The strap bonds the door to the cabinet structure. That isn’t the same thing as connecting the complete cabinet installation to protective earth.

It is entirely possible to have a beautifully bonded door attached to a cabinet whose earth stud is connected to absolutely nothing outside the enclosure.

That distinction matters: bonding connects conductive parts together; earthing provides the required connection to the wider protective-earthing system.

The little green-and-yellow strap tells you that provision has been made for bonding the door. It shouldn’t be interpreted as the cabinet announcing that the rest of the electrical installation has completed itself.