Teach children to go without. If you are well off, live below your income, but don't deny them this privilege. To go without, to have to wait or struggle for things, or live with unfulfilled desire, is one of the most valuable lessons of life for it is the condition in which most have to live most of their lives. Do not allow temper tantrums. Do not just ignore them.
Uncontrolled temper is sin.
Punish it promptly. If you ignore it, as is commonly advised, all they learn is that it doesn't work - or at least, it didn't on this occasion with this person. I know what you'll do. Thus you indulge in double standards. That is not right nor fair.