But if the OS is OEM, doesn't that mean that if you reinstall, it won't let you reuse that key?
It's been long established with Microsoft's strategy and method of linking an OEM OS to the hardware. It creates a hash-code incorporating information about essential hardware components -- mostly the motherboard.
Reinstallations to the same hardware, or hardware with processor, RAM or hard disk changes may -- or may not -- trigger a dialog from Windows that the hardware is changed. But you can simply call the automated number provided, write down the sequence of alpha-numeric characters spoken by the robotic voice on the phone, and get a fresh activation for the new install on hardware even if the motherboard has been changed out.
Two years ago, I bought an i5-3570K, Z77 motherboard and 16GB of RAM provided all ready to go. The seller sent me the white-box OEM Win7 Ultimate knitted to that hardware. I was pulling the processor to replace with another I wanted to use, and my hand slipped -- damaging the pins in the CPU socket. Trying to repair them resulted in a broken pin. So I swapped the motherboard with one of a different model and the new CPU, went through the automated phone experience, and reactivated the OS in spite of the hardware change.
Otherwise, with no hardware changes, once you install an OEM Win7 to that hardware, you can re-install and re-activate as many times as you want. You only need to re-enter the product key for each successive install.
Further, if you have "any old" Win 7 install disk of the same type ("Home," "Pro" etc.) but licensed to another system, you can use it to make the installation, enter the customer's own Win 7 product key, and still activate the OS. At least I think that's the way it works. Someone else may know with more certainty.