You have a very inquisitive child and that's a great thing in many, many ways. However, he's having trouble taking "no" for an answer (from you, from the teachers) and that's just a developmental stage.
I think there is way too much emphasis on academics for where he is developmentally right now. He's having no trouble with content, but he's not socially ready for this level of structure, nor is he quite mature enough to defer his own gratification (wanting immediate answers) for the direction of the adult or the welfare/benefit of the other kids. He doesn't yet see that it's "not all about him" or sense that others don't have the same questions. He's also apparently having problems with some of the kids and either is on the receiving end of some negative behaviors (not atypical at this age, but still unpleasant) and he's feeling rejection by playmates even if the teachers don't see it or assess it the same way. So, he's a fish out of water.
I would absolutely ask the school psychologist to sit in on the class a few times to observe him - that can give you great info and insight from someone who is only there to watch one kid, while the teacher has to deal with everyone. Ask the counselor to also attend library time and watch that interaction.
Pushing him to get through this to be ready for kindergarten is putting pressure on him, and perhaps on the teachers. I think the librarian was WAY out of line to discuss this in those words with your child present, by the way. But that doesn't mean your child is really at the readiness stage for kindergarten.
I think you have to work at home - and I'd ask the most effective teacher what language she uses to defer questions - and implement some of those at home. Yes, I know it takes longer to argue with him about why you can't answer his questions, but it's actually reinforcing the "I want the answer now" habit you want to break. What needs to happen is he needs a choice: "Do you want the questions answered, or do you want to go for ice cream? You can't have both, so choose." "Do you want your questions answered, or do you want to ride bikes? There's only time for one thing. So choose." He needs to miss out on something. From there, you "graduate" to you not having time to do X or Y because you had to spend time answering questions. That means that his favorite snacks weren't picked up at the grocery store (because there wasn't time) or something else didn't get done. It's also okay for him to see that someone else is unhappy or frustrated because of his demands. So then it's Mom's turn the next time, and there are no Q&A sessions at all - he goes to the store in his pajamas (because he refused to get dressed) or whatever. You're giving in to him because it's easier for you than not answering (which I totally understand is more time consuming). He needs to lose your attention and have you walk away. Maybe you need a code word/phrase like "This is not the time."
As always, I suggest talking to a good children's librarian about recommended books, not necessarily about school but about what happens when everyone goes their own way without thinking of how we, as a group, get from here to there. Anything visual, like a train whose cars go in different directions or a caterpillar whose segments don't cooperate, or perhaps a bunch of people with bricks who all build their own tiny building without cooperating as a group to build something larger that suits everyone's needs (something with only room for one person vs. something that holds everyone). Anything along those lines that helps him see the bigger picture, that a group cannot function if everyone in it does their own thing.
It's also possible that your child will not be ready for kindergarten in September. There's absolutely no shame in that! He is developing just fine in some areas, and is still waiting for skills to emerge in other areas. Stop pushing the area where he's accomplished, and strengthen him in the areas where he is not.
I don't entirely agree with changing schools - that teaches him that 1) he's a failure and 2) anytime things don't go his way, he doesn't have to learn and adjust, he can just leave. That's not how life works. But perhaps a play-based program is more what he needs now. Discuss with the counselor when you have more data.