Can you recite the book(s) from memory too, OP?
I remember reading somewhere (don't cite me on this though) that as we learn to read and write, we gradually lose our memorising skills. After all, we don't have to remember, we can just read it up! So we stop exercising our memory muscle, so to say, and lose some of the amazing ability some toddlers have.
From that perspective, I wouldn't push the reading so much :) Enjoy this stage while it lasts.
Age 2-3 was DS' Julia Donaldson phase. He could recite, word-perfectly, The Gruffalo, Gruffalo's Child, Room on the Broom, and A Squash and a Squeeze. He knew precisely when the Gruffalo says 'Amazing!' and when he says 'Astounding!' etc. Reciting doesn't actually do him justice - it was more of a performing thing, of diving fully into the story.
The year after, so age 3-4 was his dinosaur year. We looked at lots of dinosaur books together and went to museums etc. Eventually he knew about 100 dinosaur species, and could tell you all sorts of things about them. E.g. which period, diet, how many digits, whereabout in the world they could be found, genealogies, and famously, he'd go around explaining to people that there is no dinosaur called Pterodactyl. Since a) dinosaurs were land animals, flying relatives of the time were pterosaurs. And b) there is a species of pterosaurs called Pterodactylus. With ...us at the end. No Pterodactyl though. But many people just use Pterodactyl as a shorthand for 'flying dinosaur'. (As a 3yo he'd explain this to adults, explaining how they'd got it wrong but that it was all right as though it was not scientifically correct, it was a common misconception and popular shorthand)
What I'm saying is, maybe see if your DC is interested in applying his amazing toddler memory skills to other 'genres'? Space, trains, ... some non-fiction, find a topic that catches his/her interest? We also read some children's poetry, 'poems to perform' was a favourite book. Also songs.
DS was a good way into phonics when he started school but by no means a fluent reader. (Learned from alphablocks, teach your monster, and from reading Songbirds books with me). By end of reception, he'd cracked it, and was reading a book a day. Voracious reader until about age 9 when it got replaced by gaming... In that time of reading, picked up lots of knowledge (eg in depth Greek mythology) and also explored all sorts of stories obviously.
So yeah, no need to push the reading, it will probably happen naturally, and a few months earlier or later makes no difference.
Meanwhile explore what else your DC enjoys finding out and memorising!