When I first started in astronomy, some 40 plus years ago, I was given a cheap telescope. It just didn't do the job and I lost interest for a time, until I was given a good pair of binoculars (German WWII) to look at the stars. Since then, I never looked back (no pun intended). Boy, back in those days the skies were dark!
Anyway, this is something that I am quite passionate about. The problem with buying cheap binoculars for children is that the are not well constructed and can easily be knocked out of collimation. A child using a binocular that is slightly out of collimation may not actually realise that they are, which will affect their eyes, as the eyes try to compensate. One that is knocked completely out of collimation is totally useless. The cost of collimation also makes these types of binoculars useless, as they are simply not worth collimating.
Furthermore, if you really want to encourage a child it is best to pay for a reasonably decent binocular, one that will actually provide continued satisfaction, one that they can use throughout their lives, whether for astronomy, sports, etc.
John Ruskin (19th century) stated that "There is hardly anything in the world that some man can not make a little worse and sell a little cheaper and that people who consider price alone are this man's lawful prey".
He also went on to say; "It's unwise to pay too much, but it's unwise to pay too little. When you pay too much you lose a little money, that is all. When you pay too little you sometimes lose everything, because the thing you bought was incapable of doing the thing you bought it to do. The common law of business balance prohibits paying a little and getting a lot. It can't be done! If you deal with the lowest bidder it's well to add something for the risk you run. And if you do that, you will have enough to pay for something better."
Bottom line: you can't properly encourage a child by trying to do it cheaply; it just won't work.
Instead, buy the child a good book on astronomy. It will be more worthwhile.
Cheers Peter