For me the big turnoff of selling privately is vetting people. How do
I know that the 'viewers' aren't just burglars coming round to case
the joint? I'm not sure how much extra vetting estate agents do, but
it does feel like it's safer.

AFAIK, agents do no vetting of clients.

Estate agents *should* register applicants, and perform a money-laundering
check *before* proceeding to send details, and arrange viewings. I'll wager
not a lot do (correction, I *know* not a lot do, as their software won't
cope), but that's no excuse.

If you read the industry press, it's something they are waking up to, after
a few court cases where they have been sued (and lost) because they didn't
keep appropriate records ... I believe "compliance" is the name of the game.