Well done on your achievements so far - your English is among the highest level I've seen on here or anywhere else for quite some time.
However, my personal opinion is that if you're not brought up in a country or if you haven't lived there for a considerable time, becoming a genuine native speaker is nigh on impossible. My reasons for saying that are the same as yours - idioms, slang, phrasal verbs etc. I think a lot of people assume that we use slang in a considered way - we start speaking and wonder which bit of slang or which idiom we can throw into the next couple of sentences. We don't. It's unconscious. Some people don't do it at all. Others do it far too much. I consider myself a natural user of all parts of the language and I wouldn't think twice about saying something like:
"Ooooh, make us a brew, love. My tootsies are killing me. That boss of mine is a slave driver. Then we can have a butchers at the crossword if you fancy. I haven't got a scooby doo what's on TV tonight. Tripe, I expect."
OK, I may have exaggerated a little in that I probably wouldn't use quite so much "non-standard" English in one short paragraph but you see my point. Given that your English is at such a high level, I imagine that you can make sense of that paragraph even if you haven't seen a couple of the phrases before. But the question is, could you have formulated that paragraph?
I don't think reading children's books will help much, to be honest. If you want "real" English that contains all parts of speech, I would suggest something like watching "Trainspotting" or "Lock, Stock and Two Smoking Barrels". Perhaps "Reservoir Dogs". For me, Tarantino's genius isn't in the words he uses, it's how he puts them together and how they're delivered. His observations on human interaction are incredible. I'm sure you will easily understand what is said, but it's getting to grips with how it's said, the rhythm and the patterns that give you more clues to how native speakers converse.
It's impressive that you can understand Shakespeare and the arcane use of the language in the Bible, but if you're talking about replicating real-life everyday speech of the average person (in the UK, at least) those don't matter. You need to be able to understand EastEnders, Miss Marple, the banal tripe they trot out on X Factor. I'm not being sarcastic but neither am I suggesting that you should aim for the lowest common denominator. What I'm saying is that your grammar and English vocabulary generally seem to me to be perfect for speaking to the kind of people you might meet at a business meeting or a conference and you would absolutely be able to hold your own. If your accent isn't strong, then they may well think you're a native speaker. However, if you popped down to a street market in East London and tried to strike up a conversation with one of the market stall holders, how do you think you would do? Do you think they'd think you were one of them? Probably not. Mind you, they wouldn't think that of me either.
I have to take issue with one thing in your post. You said "Sure, they are more used to the language than I am since they've been using it for a longer time but arguably, if I put the time and work hard I should be able to equate with such uneducated natives at the spoken level."
Be careful - do not assume that people who use slang, idioms and phrasal verbs in their day-to-day speech are uneducated. That is a complete fallacy and will not win you any friends.