NOT exactly correct.
True that male bees (drones) do no pollination. But they do not mate with their own queen. Thus they fly several miles away from their own hive to mate with a queen from another hive. A queen will only have one mating flight and from that stores the sperm that will last her the rest of her life. So you can see that the chances of a particular drone being the lucky (?) guy to get to mate is unlikely. Only a few of them get lucky and then die doing it. So there are not many drones produced in the hive. The queen actually controls how many drones are made. All the other bees are female and all the females except the queen are worker bees. These workers are NOT neuter . They are true females. Sometimes if the queen of the hive dies and the hive has none of her eggs at the right stage to raise one as a new queen, then a few of the workers will mate and lay eggs as an attempt to revive the hive. These worker laid eggs are not successful though so this attempt never really works properly.
If a hive loses its queen, some of the worker bees will start to lay eggs. But because the eggs haven’t been fertilized, they hatch into drones—which does the dying hive no good.
In a healthy hive, the drones are driven out in the fall, since their only function is to mate with a new queen. A queen only mates once—she takes the drone’s reproductive bits with her—that’s why the drones die after mating. Isn’t nature wonderful?