Lol you’re right about this giving native English speakers a headache. I’m not sure the subjunctive is the correct explanation here, though.
The subjunctive mood in English primarily uses the past tense form of verbs (“were,” “were to,” etc.) to convey wishes or counterfactuality. E.g. ‘I wish you wouldn’t drink so much coffee’, or 'If I were you, I wouldn’t…"
However, ‘would you like a coffee?’ is a direct question of preference, which means it technically is using the indicative mood rather than the subjunctive. Here, ‘would’ functions as a model verb to soften the request and make it more polite.
Sorry, as per the rules of this community you’re not allowed to see them as human.