If the only goal for Finnegan is to get the most you can for him in a trade, why wouldn't you keep him in the closer role where he can show other teams his value in the that role.
It was said here Hunter wasn't ready yet for the closer role. In 6 save opportunities this year he's blown 4 of them. It's not good for a player to be put in a situation he's not ready for. Manager 101 in any sport says don't put a player in a position he's not ready for.