When I made jam from mulberries I never bothered with removing the tiny green stems. I even eat the tiny green stems. And I have no intention of removing the tiny green stems, if they are still intact, when I get to make wine.
I am currently freezing my berries in flat layers, then the single berries go into my everyday colander--for a quick shake, shake--and I have noticed that many of the stems actually come off. Then the berries get vacuum sealed and go back to the freezer.
Still harvesting every last berry I can, and seriously have to stop the "1 berry for Sara and 5 berries for the bucket" as I harvest.