tim (moved to sweden) said:

I can't quite understand what they were checking though, as
everyone they stopped had already gone through the exit barrier
with their Oyster - and they were asking to scan them. They didn't
seem to care about those, like me, with paper tickets.

As the machine keep the used tickets, what proof of payment
do they expect you to have?

The original poster clearly said they were only checking Oyster cards,
not tickets. The proof of payment would be stored in the Oyster's
records.