Legitimacy is given by the people
Right, and you know who ratified the Constitution? The People. If the authority isn't specifically listed in the Constitution, the people have not consented, and the authority is not legitimate.
No, you have your facts backwards. The people gave the constitution legitimacy. And only the people can give legitimacy. Only the original source of the constituent power can give legitimacy. Not the product of that constituent power. The constitution gives legality.
If an elected president appoints a congress representative, is that representative legitimate? No. Legitimacy is not inherited.
If the elected president makes a decision that is within his constitutional powers, it is a legitimate decision. Pay attention here: I'm talking about making the decision, not the decision itself. The ability to make the decision is what defined if it is legitimate. If the decision itself (the case here) is against the constitution, the decision is legitimate and ILLEGAL. (And overall an asshat decision, but that is besides the point).
Be careful with this chain of causality you are trying to establish there. Because, otherwise, you might end up with something like this:
People ratified the constitution; constitution establishes the election of president; president was elected; president disbanded the congress and replaced all representatives with appointees. And since it is all in the chain, it is all legitimate. So no, the making of a decision that is within his legitimate powers to make is legitimate. If that decision contradicts the constitution, it is ILLEGAL.
Remember Nixon? What he did was illegal, not illegitimate.