You may get a number of responses to this, so you don't have to pay attention to mine if you don't want to! I'm a grandmother, so I look at things from a slightly different perspective.
And, yes, it is rude to ask people to give your children money. (Actually, it is rude to ask people to give your children any sort of gift at all, since the very nature of a gift is that it is voluntary. But I know what you really mean.)
One alternative is to ask that guests not bring gifts at all. You could stress that their friendship is gift enough. Then make the parties so much fun that they're gifts in themselves, both to the birthday child and the guests.
But to say, "Bring me [bring my child] a present but make sure it's cash" - yes, it's rude, even though people do it all the time.
The savings account business is good! But start it yourself. Put a small amount of money into your child's account, and then teach your child to save a little out of every allowance, even if it's just a penny, and add it to the account (check with your bank about minimum deposits). Later, you could spread the word to the grandparents that the children now have savings accounts, and Grandma, Grandpa, and Aunt Gertrude might (voluntarily!) decide to encourage the saving habit themselves.