Grammar – Can ‘There’ in ‘There Be’ Be Replaced by an Actual Place?

I'd say it is. We still don't care about the book itself, just the act of reading it, so we can keep leaving the book "unspecified". It also reads nicer - "reading a book" is a fairly natural phrase, while "reading the book" brings attention to the book itself.

Why does that construction exist?

Because in spoken language, should have and should of are indistinguishable (when enunciated as contracted should've). And some native speakers are so ignorant they don't understand that basic syntax requires an auxiliary verb after should, not a preposition.

But note that some perfectly competent Anglophones may deliberately write the incorrect version sometimes as a facetious usage. OP's cited Facebook poster probably isn't very competent, given that "next time" is future, so the auxiliary have (part of a "Present Perfect = Past" verb form) shouldn't be there anyway. But even people who know perfectly well that it's "incorrect" sometimes write things like a whole 'nother ballgame for much the same reason.

I don't know why OP wrote it's not correct to use should have. That's exactly what the Facebook writer should of¹ written!

¹ In case it's not obvious, that's me being facetious! But actually whereas one should've = should have done something in the past, when talking about a future "next time" it's what one should do (infinitive do, not have + Past Participle done).