In the Traditional Marriage Algorithm (TMA), where boys propose and girls reject, can we say that boys get their optimal mate because boys go first and the girls wait?
Let's define ideal mate where ideal mate is the one on top of their list. With each proposal, the boys either strike the topmost girl on his list or wait, after eliminating the top girl, the topmost girl on his list after the elimination is his ideal mate. With each elimination, the boy would still get his ideal mate but girls will not get their ideal mate, because they have to choose from the boys proposing to her. Some girls might get their ideal mate but not all. Whereas every guy will get his ideal mate.
Is my explanation right or is there some logical fallacy?
It is true that the Gale-Shapley (deferred acceptance) algorithm in which men propose is men-optimal. But we must be clear what optimal means here. It does not mean that every man gets his first choice; clearly this cannot be true, for instance, if two men share the same first choice. Optimality here means that no man can be better off relative to any other stable matching. In fact, due to the lattice structure of stable matchings, this also means the men-optimal matching is women-pessimal.
You need to invoke stability in your proof of optimality. Wiki has a nice proof of optimality by contradiction based on the following picture that you can see here.