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I think that's right. Except that there may be some odd cases in which random treatment is also just.

Think for a minute - there are cases in which everyone is equal with respect to all of the relevant factors. People who buy lottery tickets, for example, are equal in all relevant respects. And we can imagine a group of people in a life boat who are equal in all relevant respects. But what if there is not enough food and half of the people have to be put out of the boat?

In cases like these a random distribution of a benefit (like winning the lottery) or a burden (like being put out of the boat) would also be just.

But these are not typical cases. In the typical case, I agree that justice logically requires attention to relevant characteristics and treating people accordingly.