the OS in question was 10.6.8, and the version they sold me to "fix" it was 10.7, just one version later.
Wait, there's something I don't get. This is the list of all Mac Mini models.
The first three use a PowerPC G4. The next three use either Intel Core Solo or Core Duo, which are 32 bit processors and therefore only upgradeable up to 10.6.8. You are clearly not talking about these, and you specifically mention that it was a Core 2 Duo.
Starting from the Mac mini "Core 2 Duo" 2.0 (Early 2009) (P7350), all the Mac Minis are upgradable all the way up to El Capitan. So you didn't have one of those.
That leaves us with only two possibilities: the Mac mini "Core 2 Duo" 1.83 (T5600) and the Mac mini "Core 2 Duo" 2.0 (T7200). And indeed those two computers are upgradable to 10.7.5. You just need to read the fine print under the asterisk:
*This system can run the last version of OS X 10.7 "Lion" if upgraded to at least 2 GB of RAM.
So there you have it. You just needed to add a little more RAM and everything would have worked. I agree that you were mis-informed by ignorant Apple Support people, but the essence of what the company told you is still true.
(That, or you are mixing up the Core Duo and the Core 2 Duo processors.)