Learn French: J'ai compris son étonnement, mais mon souhait était qu'il parte. - I understood his astonishment, but my wish was that he leave.

In modern, everyday French, the sequence of tenses is simplified. Instead of using the rare and highly formal imperfect subjunctive (qu'il partît) after a past-tense main verb, you just use the present subjunctive. As long as the action in the subjunctive clause happens at the same time as or after the main verb, the present subjunctive is correct.

When would I need to use the past subjunctive (qu'il soit parti) after a past tense like était?

You use the past subjunctive when the action in the que clause happened before the main verb. For example, if you wanted to express regret about a prior action: J'étais triste qu'il soit parti (I was sad that he had left). In our sentence, his leaving is what the speaker wants to happen next, so the present subjunctive parte is used instead.

I learned that verbs of desiring (like vouloir) trigger the subjunctive. Does the noun phrase mon souhait était que work the same way?

Yes! Nouns that express a desire, emotion, necessity, or doubt trigger the subjunctive when followed by que, just like their verb equivalents. Here, mon souhait était que... (my wish was that...) expresses a desire and acts exactly like je souhaitais que... (I wished that...), making the subjunctive mandatory.

Could I avoid the subjunctive here by using an infinitive, like mon souhait était de partir?