Firstly there would be no COA so effectively there would be no product key.
Also I don't know how these units are delivered in your neck of the woods but here they are all activated prior to delivery so the hardware/software packages are registered with M$.
The only exception to this is with Volume Licencing products which don't need activation in the first palce anyway and then there is the fact that we are susposed to supply computers with Windows loaded but no other software because the person who buys it then hasn't accepted the licence agreement from M$ so they can always refuse to accept the EULA and not get the software laoded. Howevr from my limited experience most times thay muck up the install and it requires a total reload of the product and reactivation of at least Windows.
So I guess that the answer is somewhere in the middle depending on what is in place where the computers are bought but the lack of a COA is a dead giveaway as far as M$ is concerned that you have pirate software on your computer. These are impossible to get for some reason particuarly when M$ had thousands of the things stolen in AU recently that is just the sticker not anything else.
I also had a case admittely a cheep one where the sticker just peeled off and would not stick to the case I ended up taping it inside the bottom of the case I know the wrong place but as I diodn't build the unit and I didn't want to see the COA go by by's it was the only alternative left as this was the only place without a coating on it. There must have been something in the paint that prevented these things from sticking but it did cause problems.