This sounds very familiar.
We had a rent-back seller play games with us about settlement - repeated extensions etc. They were also awful with late payment of rent and with leaving the place in a mess.
Basically, they had some fairly tangled up finances. They weren't going to clear out the whole debt with the sale - they were using part to clear some other debts and keep the rest secured by something else, and they were waiting on bank approval, or something. In other words - in over their heads and robbing Peter to pay Paul (which is why they were renting instead of moving to a new PPOR). And in the course of the lease, we became Peter who was robbed from, too.
As others have said, there are valid reasons for extending and renting back, but take it as a potential red flag. I'd give them the extension, since it doesn't hurt you, but get tough on any late rent payments ASAP. Don't give them any grace. If they are in trouble, make it hard enough for them that they screw over someone else who doesn't push back as much before they start screwing you.