For the record, I never argued for this, I'm not sure why you are assuming I did.
"You can go with equality of assistance, and give them all 4 apples, so that now Tim has 5, but John now has 7, and Mary who already had as many as she needed, now has a huge excess at 9."
I agree that that is silly. If we can agree as a society, by vote or legislation, that paying for students lunches if they cannot afford it is worthwhile (and for the record, as a non-parent, I think it is) then we would not have any reason to give any to Mary. I would agree that if they each need 5, they should each get a share relative to how much the program can provide given its funded resources. Since these programs tend to be strapped for financial resources, giving any to Mary is wholly unfair because it probably means failing to meet minimum requirements for another child.
It's a lot easier to just state "I am for equality of opportunity, not equality of outcome." That appears consistent with your second example, and it has always been my position going back to when I was still a child.
Equality of outcome is both impossible, and textbook Communism. Equality of opportunity is perhaps also technically impossible, but I think it's more plausible and clearly causes far fewer negative externalities.
I think the levels of the analogy are getting messy. equal opportunity is, in my analogy, each student having the 5 apples. What they do with the energy and nutrition from the 5 apples is variable, but getting them all to the even playing field of 5 apples is the goal. And the law (the law in this case being the rules in place to get them all to equal opportunity) simply cannot treat them all equally as they are not equal, the needed input to get them all to the same opportunity is simply not equal. Haveing the Laws of "apple stimulus" treat them equally would be foolish.
So lets drop the analogy and go back to the real world, to get to an even playing field, to have an equal chance at life, liberty, and the pursuit of happiness for everyone, regardless of the demographic they occupy, demands that the law address that demographic inequity. And just like with the apples and the students, giving the same amount of civil rights legal help to Latinos, Black Folk, Gays, Straits, White Folk, etc makes no more sense than giving all of the students the same number of apples regardless of how many they started with.