It just seems frivolous, dilatory, and procrastinative to wait for children to go to school to learn a skill that extends much further than academia does. The skill is necessary for academia, profession, daily life, and even free time just for fun or family issues.
I want to challenge this claim of yours. Children develop. Is it frivolous, dilatory, and procrastinative to fail to teach a 4 month-old to walk? Talk about a skill necessary for daily living! Of course not, a 4-month old is not at a stage where they can learn to walk, they will reach that stage much later. In fact I think most people would agree you don't really teach a child to walk at all; they learn it on their own at a certain developmental level given the opportunity to do so. Modern child development experts would even argue that many attempts to try and teach them to walk (with those walker toys for example) hurt more than they help.
Of course not all childhood abilities fall in the same category - a child won't spontaneously learn calculus unless they specifically set their mind to it for example. So the question would be where reading falls, and my understanding is that "school age" is comfortably within the appropriate developmental window for children to learn this skill. Many children learn it earlier, but many children also suffer for being forced into learning it earlier.
This is relevant to your question because you aren't just asking about a child being able to parse words - functional literacy is about understanding what's written... But written works contain a variety of very different concepts, not all of which are easily understood by very young children! Just like you wouldn't expect a 7 year-old to understand a treatise on calculus, even if they could parse all the words and letters in it, it might not be trivial for a 4 year-old to understand complex causal relationships, the internal beliefs of characters who aren't them, and such things that are common in the works you might want them to read. 4 year-olds are on the cusp of developing abilities like knowing that what's in their head is different from what's in other people's heads, or knowing a narrow glass doesn't contain more water just because the water goes higher.
I'm reminded of a passage in Daniel Pennac's "The Rights of the Reader", which is not about child development and is very opinionated in a way that disagrees with you (but might still be an interesting read for you; it's about how to instill the love of reading in children). It talks about the transition from a small child being read stories, and then once the child is learning to read suddenly the challenges begin - "Why did Snow White fall ill? Come on now, it's easy! The apple. It's the apple. Read again!", but when the child was just being read fairy tales at bedtime nobody asked them why Snow White fell ill, and in fact they probably didn't get it had anything to do with the apple at first, but it didn't matter then. And now suddenly it does, and reading has become this minefield-ridden chore where stories used to be a joy and a refuge.
So this is all to argue in a similar direction from other answers, but just giving a more detailed rationale. I think for one thing you should read up on childhood development, to have a clearer idea of what you can expect from a given stage and what things you can hope will be learnt on their own, and which things you need to actively teach, and which things will be learnt on their own but you can still promote that learning. But read, read, read to your children, and ask them questions about what you just read to test their comprehension, and I would argue this: only do it to test their comprehension, for you to know where they are at, or to model how you yourself relate to stories ("Wow, brown bear hit that other bear! I think he must have been angry"). Not as a test that they need to provide the right answer to. Trust that if they give wrong answers at first, it's just them being too young or too inexperienced with the structure of stories, and that they'll get there, but that you just got valuable information on where they are at. And also, that just by asking the question you are already exercising a part of their brain that seeks the answer, so you probably are helping them develop functional literacy in time.
And I would add, please prioritize your relationship, especially in the younger years. If they seem upset by your questions, maybe lay off them or find a different way of asking them. Your children will learn better over the long term if they feel safe and heard by their parents, if they think their needs and wants are being hearkened.
From other comments it sounds like you have pretty technical concerns about what questions to ask, what information you should prompt them to derive from their reading. I have less opinionated answers on that, but I can come up with ideas. For example, maybe at the earliest stages just do basic recognition of objects - "what's this? A ball! That's the bear!". Maybe more complex would be recognizing actions ("What's the little pig doing? It's kissing its daddy!"). Later, questions about the internal processes of characters ("The little pig is smiling! How do you think he feels?") or basic causality ("The shark seems mad! Why do you think that is?" or "Look, the ball fell on his head! Where did this come from? You're right, it's the ball she dropped on the previous page!"). Consider, what situations might your child find themselves into, what challenges might they face - problems sharing, problems dealing with sadness or anger, arguments with friends... children's books often involve messages about these things, once your child seems adept at understanding what characters are feeling and why, consider questions that go to the "point" of the story ("Little brown bear was mad and didn't want to share! What happened afterwards? Do you think it's good to share? Why was he happy at the end?"); almost by the definition of a children's book they will be things that are relevant to your child's life. You could also with all those things model the behavior your soliciting by giving your own thoughts, or making comments about your own thoughts instead of asking them ("Shark is mad! Wow, I'd be mad too if the octopus had stolen my ball").
When they're older they'll have to do all this at school anyway, I think you can probably take your foot off the pedal and see what kinds of questions they get asked at school, how adept they seem to be in answering them, and see if what they need help with if anything. Maybe something fun you could do is assume that there's a certain type of question they're asked in school (questions that are about the theme or message of the text, usually) and challenge them with very different, silly questions that still require interpreting the text, such that they'd be exercising slightly different mental muscles than in school but in a fun way that puts the school questions in perspective. Like, who would win in a fight, this or that absurd combination, or what would have happened if these characters had done something silly instead of the tragic action they did take...
I think with all these things it's important to model the behavior you wish. Do you have yourself an interest in interpreting texts, understanding the intent of an author, seeing what morals you can derive from a story and whether you agree with them, that kind of thing? If so, just share that with your kids. If not, maybe try and develop those skills yourself so you get a deeper understanding of how they work, and you'll probably find it much easier to transmit them then.
