How can I effectively lock down an iOS device for multi-app use for children?
You can use "Apple Configurator 2" from the App Store to supervise the device. Note that supervising the iOS device will wipe its current content.

When you have supervised the device, you can setup restrictions such as for example disabling the Safari web browser, disabling app installation from the App Store, etc.

Another approach might be to make an agreement with your children so that they know what they're allowed and not allowed to do, and then trust them to not install other apps. If they do, you can have a productive conversation on that topic - perhaps they can convince you that the new app is actually a benefit to them...