Tense in subjunctive sentence

Most of that is irrelevant. What's relevant is the following noun phrase (NP), an entire sentence serving as the object of the preposition "than":

"they would have if they had stayed together"

That's a complete sentence that's serving as the object of the preposition, so let's write it as one:

"They would have if they had stayed together."

That could conversely be written as:

If they had stayed together, (then) they would have.

No tense is being employed. This is an irrealis, a conditional sentence in which the main then-clause employs the past conditional mood "would have" and the subordinate if-clause employs the pluperfect subjunctive mood, these two moods being used to talk about an unreal situation in the past, a hypothetical scenario in the past resulting in a conditional outcome in the past.

If the unreal situation weren't in the past, it would be written as:

They would if they stayed together.

That could conversely be written:

If they stayed together, (then) they would.

No tense is being employed in that, either. Just like before, this is an irrealis. The difference is, rather than use the pluperfect subjunctive mood and the past conditional mood, it uses the past subjunctive mood "stayed" and the conditional mood "would."

By the way, "past subjective" is a bit of a misnomer in English since its use in English is to convey unreal situations that are not anchored in any specific time. Since the situation is unreal, it actually has no time. This is as opposed to the prior example that uses the pluperfect subjunctive to anchor an unreal situation in the past, pluperfect subjunctive being different than the past subjunctive since the pluperfect subjunctive actually is anchored in the past.

You say:

The tail end of the sentence, from a news article, after than describes a contrary-to-fact situation in the past tense (not about the verb form) as the divorce is already set in motion. The author apparently sees this section as a past event, a perspective that I cannot wrap my head around.

Here's how you wrap your head around it:

They are already not together, so the contrary-to-fact situation in the past, as you put it, is the unreal past action of staying together, the not-contrary-to-fact situation in the past being the real past action of not staying together.

Since the real past action not staying together is in the past, its alternate unreal past action staying together, the unreal action that didn't happen because that real past action did, is likewise an unreal action in the past, which is what the pluperfect subjunctive mood "had stayed" and the past conditional mood would have is used to convey, their raison d'être.

You also say:

But I wonder whether it is possible to describe the situation in the present tense--they would if they stayed together. Indeed, I feel the present tense is more appropriate in this subjunctive sentence.

Again, to be clear, there is no present tense used in that. It's all past subjunctive mood and conditional mood. As I explained above, that clause could be expressed as you wrote it (i.e., "they would if they stayed together"), but not in the given situation where they are already apart in the present and it's presented as an unreal alternative to the real past action of not staying together. That means what you feel is more appropriate in this subjunctive sentence is not more appropriate or appropriate at all.

You ask:

Could you please explain why the past tense is applied here?

It's not. The pluperfect subjunctive mood and past conditional mood are applied here because it is presenting an unreal hypothetical scenario and outcome of them staying together in the past, which they did not do in the past and so cannot do in the present because to stay together in the present, in the sense conveyed of remain together, would mean they are together in the present and are not not together in the present when we know they are not together in the present as the action of not staying together is the real past action that the unreal past action staying together (i.e., had stayed together) is meant to replace, like in some alternate reality where they had stayed together in the past instead of splitting up in the past.

If something is stopped in the past and not restarted, then you cannot aptly use the verb stay in the present to refer to it being perpetuated as that would contradictorily mean that it either was not stopped or that it was restarted such that it is the current situation you convey you want to perpetuate with "stay."