My 3 year old (almost four now) is like this.
What I’ve noticed:
While he still loves all of his old shorter books too, he is perfectly able to handle longer books with chapters (such as Winnie the Pooh, or Roald Dahl books like James and the Giant Peach or Fantastic Mr. Fox). However he usually needs to hear the same story two or three times before he really gets it, and he likes to hear the story probably 30 times before he gets tired of it. We have read all of Fantastic Mr. Fox nearly every day for the last month. He now likes to recite the book to himself while he’s playing.
Like you, I also don’t want to push it, so I usually ask him what he wants to read. Sometimes this means short board books which we have been reading since he was born, sometimes it’s longer chapter books. However, he obviously only knows of the books he has already read, so it is my job to introduce him to something new. When I find a new book I would like to read with him, I say “mommy wants to read this one tonight for a little bit, but then we’ll read two books that you pick out,” and he’s usually pretty happy with that. I will read a few (maybe 10) pages and keep an eye out to see when he stops paying attention. Usually he won’t like the book much the first time, but if I wait a week or so, then he will ask for it because he remembers the little bit we read and wants to see what happens next.
If it was too advanced for him, he doesn’t ask to continue with it. This has happened with Harry Potter, Ralph S. Mouse (though he liked the first one - the mouse on the motorcycle), and Rascal. Those books have been shelved and we’ll come back to them in a few months and try again. But he’s enjoyed all of Roald Dahl except The Witches and Matilda, as well as Winnie the Pooh, Alice in Wonderland, The Dragon in the Sock Drawer, and even How to Train Your Dragon (though with that one he frequently stops me to ask why the characters are doing things like banishing their children etc).
Ps - something that really helps him engage with a book the first time through is if I read each character with its own exaggerated voice and mannerisms.