I second the Kobo recommendation. Bought a kindle for son, since the "kids mode" was supposedly better. But I absolute hate the kindle! It's not just about the ads, also the constant upselling and awful UI formatted around selling you more books! For example the "homescreen" is a small sliver with a couple of your own books, then several rows of "recommended" books to purchase! And forcing you to scroll to next page to get new/added books! Disgusting. I didn't realize how they could make a simple book UI this terrible, until I used the kindle. The kobo UI is about showing your books
Try using the Library screen. It shows only your books. You don't ever have to use the Home screen if you don't want to.
Just checked this, and maybe just the "kids mode", but actually no. Kids don't have a Library view, only home (aak "upsell view) and "recent". And recent has all books that have been read within a unknown time, and no way to remove them, even if deleted!

My son is constantly asking me to remove books he's finished, but they still stay on recent view, clogging it up. Aka: the only way to view your library, sort of.
Oh, and Amazon won't let me add public domain books to the kids mode. I have to approve content, but those never even show up. But hey, you can always buy them.. So yeah, fuck Amazon. If this is the best eReader for kids, with parental controls, then it's really terrible. FWIW kobo removed their kids mode, and people report the parental controls there are garbage too.