Tense – Using Correct Tense After ‘After’ in a Future Context Within a Past Sentence

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.

As @snailboat said in comments:

Shifting to the present tense in the middle of a narrative is a use of the so-called "historical present". It has the effect of making a story more vivid, more immediate. I think if you search the site for that term you'll be able to find some more detailed descriptions of it.

Most stories are set in the past, and as such are properly spoken in past tense. However, speaking of the past in present tense can indicate that the speaker's recollection is so vivid that the speaker almost feels transported back to the moment of the events. Speaking in present tense is also common trick in storytelling to a create greater sense of immediacy for the listener, in hopes that the listener will imagine the events in present tense, as if the listener were a witness to the events as they are happening.