I think the skill can be quite useful in some circumstances. While the non-proficient skill check might seem to be easy, a DC 10 when your life is on the line can make you want every plus to the check possible. So a +2 or +4 to the proficiency can matter.
The biggest thwart to the usefulness are healing kits. However, there is a variant (I think in the DMG) that says you cannot stabilize anyone without a healing kit. In other words, the healing kit allows you to make a Medicine check to stabilize. There....now the skill is quite useful.
I would probably rule that you could try it without the kit, but at a disadvantage. Now the skill is even more useful.
Also, Medicine has been used in my games to figure out what killed someone (bite, claw, sword).