I think this is a matter of opinion; but you come close with In fact, not every person using the expression would be aware of its provenance.
It becomes an "expression" or "colloquialism" when it starts being used as a figure of speech by people that heard it (obviously) but have no idea where it came from, or who authored it, and just don't care.
In order for that to happen, the phrase needs utility, it has to apply to enough situations that it is worth remembering. It must itself be memorable, a clever turn of words or sounds or unusual words, so it feels witty or erudite to the speaker. In your example, "oft go awry" is poetic, and sounds both "Old English" and formal and erudite. It sounds like you are quoting some famous writer. I can't explain why, but it is fun to say. (The whole thing, imagining mice making plans is fun.)
The "burned hand" bit is nice enough, worthy of a quote and applies in enough situations, but IMO it is not up to the mark of the "oft go awry" line.
The more insulting "Those who can't do, teach" line makes the grade, due to the surprise ending.
In this respect, I think the distinction is subjective, and this is much like joke writing. It can be difficult to be witty without being wordy and to deliver the punchline in the last word, to encourage the reader's mind in one direction and pull out the reversal punchline in the last word or two. But we are seeing something very similar going on here; the "best laid plans of mice and men" is the setup, there is alliteration, there is a suggestion of careful planning, and the equivalence of the planning of mice and men, and the "oft go " doesn't give away the surprise punchline: "awry".
That same kind of reversal doesn't work in "The Burned Hand Teaches Best"; the negative is given up front, and reversing to a positive doesn't have the same impact as reversing to a negative. A slightly better version would be "The Hand that teaches best is scarred," or going off script from that, "There Is No Teacher More Thorough Than Failure."
In a way, this IS joke writing, the phrases that make it past the "quote" stage are basically one liner jokes that really do apply to enough life situations to be worth remembering. They make people laugh, smile, or snort, or roll their eyes. As they say, it's funnier when it's true.