Tense – Using ‘Recently’ in Present Tense

This is a bit clearer if rather than "still relevant" you think of the perfect tenses as expressing a current (with respect to Reference time) state, not an event.

The English phrasal verb be born, however, designates an event: birth, which is the punctiliar fact of entry into a state of extra-maternal existence (I am being as pompous and mealy-mouthed as possible to avoid politico-theological controversy). An event is expressed using a simple form, not a perfect form.

There are circumstances in which a perfect construction with born is proper; for instance:

A son to the Earl has now been born, so his nephew Roderigo is no longer the heir apparent to the title.

Here a current state is spoken of: the existence of an heir presumptive. But if you want to locate the birth in time you are speaking of the event, not the consequent state, and you must employ the simple tense:

A son to the Earl was born yesterday, so his nephew Roderigo is no longer the heir apparent to the title.

Your book is correct. You've misunderstood the application of the rule.

The structure here is [ "before" + past perfect ], and presumably there is also a main clause in the simple past. This means "before" must come directly before the past perfect clause and subordinate it.

You give two examples where you believe the rule has been violated, but neither of those sentences has the structure [ "before" + past perfect ]:

All the leaves had fallen from the trees before we arrived
Before I had a chance to thank him, he’d gone.

Both sentences have the structure [ "before" + simple past ], so this rule does not apply here at all.

The general rule that always applies with "before" is this: if you have two clauses, and one of them is subordinated by "before", the other one always happens first. Where there's an apparent conflict, the word "before" takes priority over the verb tenses.