It has been 2 years since I played Oblivion last time, but IIRC Oblivion can't harvest the flora like you described(You need Harvest Flora to do that).
Anyways, I seconded Graphic Herbalism, or you can look http://www.mwmythicmods.com/Gluby/Gluby_Herbalism_Mods.htm for any other herbalism mod.
Actually, Oblivion can harvest flora like I described, since I asked for a mod that allowed me to harvest plants like it was done in Oblivion, sort of by definition.
I wonder, did these mods spring up after Oblivion came out? Or were these mods out before Oblivion, and Bethesda decided to integrate them because they saw the practicality of it?
"As in most herbalism mods, you pick plants just by activating them instead of opening them like a container." This is basically what i was after, but this part, "But in this mod only the part you pick disappears instead of the whole plant..." concerns me. Does the plant ever come back?