As you mentioned in your first sentence, all such labels are artificial shortcuts. They are tools.
But, if you understand "subjunctive" as meaning (in part) using the so-called 'past' tense in #1 distancing the 'having' in time, and using the past tense for hypothetical events, then you can refer to it as "subjunctive" without having to define everything (the distancing role, the hypothetical nature of the proposition, etc).
If a student is introduced to the meaning of "subjunctive" as it is used in other languages, and arguably in English, I imagine it would be much easier to respond to questions by saying "It's subjunctive" rather than by explaining all the properties of subjunctivity each time and attributing those properties to the example.
I can certainly accept that the labelling of constructions as "subjunctive" is almost dead in British pedagogy. But I'm not convinced that the use of the past subjunctive (under a different label) has changed much.