[kmm]

I don't think that's very plausible. I believe children don't consciously think of the language they're learning, let alone about linguistic abstractions like gender, tense, case,... The concept of a correction doesn't occur to them, in my experience when children are corrected they don't understand that the sentence fragment uttered applied to what they said. Besides, in many cultures children's speech is never corrected, and they all turn out speaking fine at the same rate.

The example you give is a bit ironic too, the linguistic shift from "X and I" to "me and X" happened centuries ago and even though it's become a classical example of an "error", there is little hope in ever reversing it. It's always been curious to me that this "error" is pedantically pointed out, but nobody cares about the same "mistake" being made in "Who wants cake? Me!", "He is older than me", "They are going to the beach, but me, I'm staying here" and of course "It's me". By the same reasoning of applying case, in all these sentences "me" should be "I", and especially the last one should be "It am I" (as it is in German and Dutch, translated word-for-word).