grammaticality - has just had or just had a baby

My question is not whether the correct grammar is either "She has just had a baby" or "She just had a baby".

I am aware that the official grammar is "She has just had a baby".
But in a way that sentence sounds wrong.
It is correct in the meaning that it follows the grammar rules and guidelines.
However, when you "have just had a baby", somewhere in my mind it is as if that person is still having that baby. As in, the birth is still going on.
But when you say "She just had a baby", the birth is done. It's finished.
Which is (hopefully) the case.

So my question is actually more of a wiki-question expecting opinions on the implementation rather than a question expecting a valid and factual answer.

Answer

She has just had a baby is absolutely idiomatic in British English.

I observe that She just had a baby seems to be idiomatic in American English (and has been creeping into British use over the last few decades); I think she has just is also idiomatic in American usage, but I'm not so sure of that.

Your understanding of the perfect is not quite right (unsurprising, as it's very subtle). You're right that it means the speaker is thinking of the action as having present relevance; but it does not imply that she is still having it.
It's more that the just brings the event so close to the present. I would hardly ever use the simple past with just - in my idiolect they are inconsistent.

[Of course there is a different meaning of just which I do use with the simple past, just meaning something like only, as in I just touched it and it fell!.]

And by the way, there is no such thing as "the official grammar".