rofl!
you can reinstall as many times as you want, i have reinstalled windows 7 and activated it, same for windows xp so many times on the same machine
you don't need a new key, if microsoft starts to do what you say, then you need a license each time a big virus messes up your system, in that order of ideas i would have to buy windows 7 around 7 times for a laptop we had in the house for 5 years
now moved to windows 10, what, buy more keys for digital rights?
technically you are not saying nothing that happens in reality
if you have a coa, you can activate
the thing with some hp coas is that they use a restore image, especially on those old win7 , xp and vista era machines, the image used a generic key, something like a volume license key
when i had to reinstall the pc without the image i discovered tht the key in the coa didn't worked, why? because i had to call microsoft to do a assisted activation, the key was never used before, only the preactivated install on the restore image was activated
but that activation was a one time only procedure
later reinstall didn't asked for activation, just put your 25 letters and numbers and that's it, windows is activated
i did that 3 times in a year thanks to problems in hard disk and other stuff that happened, always got activated, that was a coa on a lenovo laptop, a oem coa, iirc it was a windows 7 starter, or a windows vista, can't remember atm
i did something similar with a hp i used to fix for a neighbor, clean hard disk image, wipe partition, install windows for the coa, call microsoft and activate the license, it does work