One could make an argument for welcoming the end of newspapers, especially the tabloids, as far as the royals are concerned. However, the Internet, 'news' sites, forums, blogs etc., aren't exactly glowing in their coverage of the royals either. And, unlike newspapers they aren't bound by any libel laws or editorial decisions. On the Internet people can and often do say exactly what they think and facts go out the window in a way even the tabloids don't attempt.
I give you, for example, the horrendous and totally disgusting speculation about surrogacy and Kate's pregnancies which goes on unchecked in another forum day after day. God help any naive young person who knows little of the royals reading that tripe.
People reading newspapers can at least see photos of the royals going about their business on various occasions even if they can't be bothered to read the story. With papers gone what proportion of the British people are going to look up various royals on the Internet if they aren't particularly interested?
Most will then only see the royals occasionally on the TV news or on documentaries many won't bother to watch. (Free to air TV is dying as well.) From there it's only a tiny jump to "Well, we hardly see them anyway, so what use are they?"
When newspapers go I shall be sad for all sorts of reasons. However, I don't think anyone should fool themselves that this is going to herald some bright new dawn for the BRF.