[thaumasiotes]

> I think that none of these require historically correct Latin pronunciation.

That is true, but there are questions for which you want to know what the historical pronunciation was. For example, it's much easier to make sense of ancient misspellings if you can recognize that the misspelling and the target word would have sounded the same. Something similar holds for ancient jokes.

> I assume that at the Vatican they speak some derivative of mediaeval Latin

This is vague; the syntax and pronunciation of Latin change continuously into Italian. The rest of Italy has a better claim to be speaking "some derivative of medieval Latin" than the Vatican does; the Vatican's version presumably reflects mostly-arrested development after a certain point, as far as syntax goes.

As for pronunciation, I tend to assume (without any relevant knowledge) that the Vatican's Latin pronunciation is just how the same written words would be pronounced in standard Italian. You can find similar conventions for other dead languages -- it's the only option available for Chinese where we just plain don't know what the pronunciation was in 600 BC, and it's "Reuchlinian pronunciation" in Greek. This approach has quite a bit to recommend it, most prominently that it's easy to find fluent models of the pronunciation of a living language.