Never read them but yes that's cringey! I tend to read thrillera and every so often they will promote other writers books, I assume it's a bit quid pro quo, so the lead detective on the case takes a bit of time out to read the new James letters on book. That annoys me too.
To be fair I never took it as self promotion more like a funny little quirk that kids would enjoy. Never seen it in adult books and like you say Enid Blyton and Jacqueline Wilson are/we're doing pretty well. It was hardly desperate desire to sell more books
As a child I quite liked it in the Enid Blyton's as you thought "oh yes, I've read that too." It made them feel more real, and I can only remember occasional references. I think the Secret Seven have a point when they're bringing books and one of them says "I'll bring my famous Fives they're wizard," or something similar. I don't recall any other references.
If they start being promotion or too frequently then I would start to become irritated. I can remember having an (American) self help type book that was full of "other authors say that in this situation you can do X Y Z. I don't agree with that. Don't do it. The only proper way to sort it is listed in my book <insert title> IBSM 17214637854 which can be ordered through my website along with lots of other merchandise www.youlotaresuckersifyoufallforthis.com..."
I've certainly seen authors reference other books in children's books (eg In PunchBowl Midnight, Andrea thumps down a copy of "We didn't mean to go to sea" and proclaimed everyone in it was more sensible than Peter") and I think it's nice little touch to use a book the children have heard of. It probably does make them more likely to try and read them, but I don't think it's put in for that reason.