I have never understood why "Received Pronunciation", "Queen's English" and "BBC English" are all often treated synonymously.

Voices on the BBC (of today) normally sound to me rather neutral and "standard".  If someone on the street spoke to me in a similar manner I probably wouldn't even think about it twice.

The way the Queen/Royal Family speaks is to me still very stiff and posh-sounding.  Even though the "poshness" has dramatically decreased over the last couple of decades, very very few people speak like that and I would be pretty shocked if I met someone on the street with that sort of accent.

There was most certainly a time when BBC English and Queen's English were pretty similar but not today.  Both have come down in "posh-ness" but the BBC's by far more.

I'm keen to know what others' perception of RP is in the modern English-speaking world!

Yes, it's probably not the be all and end all. As for English learners, I'd estimate that they are better off concerning themselves with connected speech than speaking like Prince Charles and the like. Unfortunately, far too many get embroiled with accent reduction to a point where it takes over communication - I feel.

We Americans seldom concern ourselves with RP. We hear BBC English and we understand it easily. Most of us elect to ignore the difference as we would with any other regional English. 

For me, it's the standard accent. Well-educated people like professors, in addition to what you've mentioned, use it.