and that doesn't happen, so I don't know what you are talking about, at least it never happens on the games I'm watching and I watch quite a few of them.
The "Save" stat was watered down, I think, around 1969.
Some things about baseball are still evolving - a sea change (equivalent to the extinction of the dinosaurs) was Tony La Russa developing the late-inning specialist-centered bullpen.
The 'pen used to be just five guys who couldn't throw 300 solid innings a year. Look at the now-seemingly ridiculous workloads pitchers had to toil under pre-1968. Think of how much longer careers could have been extended (e.g. a Mel Stottlemyre or Warren Spahn). Everyone has a "closer" these days. Most if it, I think, is because of the large guaranteed contracts given to players: creates economic incentive to avoid excessive wear-n-tear, and that shows up on the field.