The 1N4742 is a 1w part which at 12 volts is well within the amperage
ratings of a 7812.
Come again? The 7812 is a 1A output regulator. So the load current can
vary between 0 and 1A. If you replace it with a resistor + zener circuit,
the zener current would haev to be around 1A when the load current was
essentially zero. That means the poor old zener is dissipating 12W
Now I don't believe for an instant that the character generator chip
draws anything like 1A, so you could make do with a larger resistor and
thus less power dissipated in the zener.
-tony