Welcome to the wonderful worlds of both Android and Android Forums, Stephen Dow!
What I'm gleaning from your post is that you're not eager to let Google, facebook, etc., have access to your personal info. This is very understandable and, especially in the case of facebook, very wise.
However, using an Android [Google's product] device without having a Google account is going to leave you with very limited options. I have no experience with Lenovo, so I can't address their parental controls, but Android and Google's, combined, are very good. Then throw in limiting the tablet's ability to go online, and you'd be in pretty good shape.
Yes, Google definitely collects your data, but you're given great control over what they can and can't do.
There are several problems I can see with your desired scheme. One being a matter of space--downloading everything you want the kids to have access to will use up a lot of space. Two, some things require an Internet connection for full functionality. Some games let you sync with their server, so game progress is preserved, and you can pick up where you left off, even on a different device. There's more, but that'll do for now.
I think that creating a Google account and then going through its settings with a fine-tooth comb should be workable for you. Then set all the parental controls available to you, including disallowing purchasing apps without a password, disallowing the device going online by making it require entering your Wi-Fi network's password, etc., and you're good to go.
Do "launchers" have the ability to set up separate user accounts with different permissions?
No. Launchers just give you cosmetic control over how your device looks/acts. If you're interested in seeing some things my launcher can do, please read
this thread about ADW Launcher.
I know you can create different user accounts on your device, presumably with different permissions. Though I've done this for decades on Linux proper, I have no experience or need to do so on my Android devices, so I can't help much there.