Ref the string on the collar to tie the dog up - I have found a few lost dogs in my time. The important thing is to get hold of their collar as quickly as you can and get some sort of way of stopping them running off. To this end in the past I have used scarves, shoe laces or dog leads (in the days I had dogs).
If you're even two feet away from the lead on the floor when you manage to get the dog (and they can be slippery little suckers!) then you wouldn't want to faff around trying to get the lead while still holding the dog by the collar (and you might not spot it until after you've got the dog sorted). So, if I had something useful about my person, I'd just use that. I don't tend to have string I don't think, but some people might, especially if they have a small holding.
I guess I might then put the lead on, but there's be no way to attach most leads to a bench/something to stop the doc wandering off, so again, string might be more useful.
I don't find the idea that someone used string implausible. I do find it implausible that someone said maybe she had a medical episode and slipped and then they walked off and left everything as it was without alerting anyone*.
But, as we know, a lot of this can be down to very lax and inaccurate reporting, not helped by imprecise police statements.
*But as someone above said, people don't want to get involved. I once gave a lift two miles, on Christmas day, to two lost women who couldn't find their car, my then bf never stopped going on about it because he nearly missed the time the pub shut and had to phone ahead to get someone to order for him. I was astonished that he would have just left them there. But that's people for you!