Just curious what things look like if you go this route, as I've always thought that most book sales for kids come from buying physical copies in stores or grabbing them from libraries.
I think it's very difficult to self publish anything younger than YA personally, though people have done it. Self publishing in general can become a full time job, but with kidlit I think it's even more so as you are 100% correct: kidlit requires physical copies and needs to get in the hands of librarians, teachers and on the shelves of bookstores. Anything of course is possible if you put in the work yourself, but I do think it's a much harder slog to self publish kidlit over YA and Adult. If you are considering self publishing I highly recommend school visits as a way to self promote, which again is a far more time consuming and costly way to promote than, say, a blog tour. (Fun Fact: All the usual avenues for promotion tend not to work for MG. Yes there are some librarian specific blogs and stuff, but all the wonderful opportunities offered YA books just don't exist for MG. That's another challenge. As someone who has done both YA and MG it is stunning how much easier it is to promote YA etc.)
Another issue with self-publishing kidlit (including YA) that ties into the library sales issue is that libraries tend to make their buying decisions based on book reviews in places like Booklist and VOYA and Kirkus etc., and it's close to impossible to get a self-published book reviewed in those outlets. I like being a hybrid author with feet in both trade and self-publishing, but I keep my adult books for the self-publishing side and let my agent submit my YA.
However, one correction--small press can and certainly do produce physical, paper books--but they're POD, not off-set print and warehoused. And some stores are squeamish about ordering POD books from CreateSpace, which is an Amazon company, so it's best to make a print book available through Ingram...which is another set of hoops to jump through that many small pubs might not have the bandwidth for. And yes, I like mixing metaphors.
Good luck with whatever you decide.