To be honest, I can't claim to understand why Bugsy Malone is enacted entirely by children. To be even more honest, I also don't understand the appeal of having a child-acted movie sung by adults. It's definitely creepy, whether intentionally or not.
The class struggles, while addressed, aren't necessarily a main theme in the film. The plight of the janitor, who wants to be a dancer, is dramatized in a brief song and dance number. However, it never comes back, and he never gets to realize his dream-- he simply remains complacent with the status quo and fades into the background as the gangsters descend into chaos.
The Depression Era economic hardships are only mentioned once, and they're coated with the same candy glaze as everything else. When Bugsy and Sam are looking for new talent, they're able to find willing and able men (boys) at the soup kitchen. After the hungry sing and tap dance about the condition on the table (knocking off their soup and bread), they proceed to be eagerly recruited by gangsters. Now, this could be seen as poor plot development, or as some sort of social commentary on how crime is bred by necessity. The second one is more interesting and gives more credit to the writer, so let's go with that one.
Meanwhile, the individual that capitalizes on human desires and the weakness of others is, in the end, "splurged" like everybody else. However, he or she is, according to the optimistic refrain at the end, just as valuable a human being and completely redeemable. His or her (mostly his) actions in this world were driven by the same need as everyone else's were, and his wrongdoings were simply a result of being born to the wrong circumstances never having access to support, love, or understanding. In this, the fact that all the characters are children becomes important. They're all children, and this entire movie was a play. They didn't die, but adults who act like them do. They, like the future of humankind, have hope if they learn how to treat each other well and how to satisfy their own needs without hurting others.
Of course, there's still the issue of the singing voices being dubbed by adults. I thought that this was incredibly jarring, because most of the dubbing jobs seemed only partly synchronized with the images on the screen. Whether this was because of a bold philosophical choice, or because they weren't able to find child actors who could sing, I don't know. While I give credit to the writers for the social commentary, I can't say that the singing dub-job had any thematic effects. Even the director and composer later doubted this choice, which was more of a production-related decision than an artistic one. However, if one wants to be incredibly generous to the creators of Bugsy Malone, one can argue that the use of adult voices was a method for tying the action in the film, acted out by children, to the adult world.
Bugsy Malone is a unique film. But whether by choice or by accident, the effect of the movie's casting choices and musical numbers is one that leaves you thinking: "Whaaaaaaaaaaaaat?"