I'm not going to read through all the interim pages of posts here, since this thread (and the one at the other site) appears to have become needlessly long, but on the first page they mentioned not using the computer for gaming, and it doesn't sound like they typically use it for demanding workloads in general.
So, in my opinion, their existing cooling solution is probably "fine". There doesn't really seem to be a problem here, at least not anything that is likely to affect the processor's health. Today's CPUs are designed to run at higher temperature levels under load. Intel lists the standard operating range for the 12700F as being up to 100C, and the processor will automatically throttle performance to avoid exceeding that temperature. I could see one not wanting their CPU to run close to that limit frequently, especially if the throttling were to noticeably affect performance, but it's arguably not going to be a problem if they occasionally see temperatures in the 80s when running a demanding task like a virus scan that may be operating on all cores. During more typical workloads, most of the processor cores will be sitting unused, and temperatures should be lower.
Now, while a better cooler isn't necessarily required, one might want to get one if they find the noise levels of the system to be distracting with the stock cooler ramping up to full speed. It probably doesn't need to be a high-end cooler though, considering the system doesn't seem to be getting used for demanding all-core workloads all that often. Something like the Pure Rock 2 that the shop suggested would likely be plenty to keep the processor at decent temperatures at moderate fan speeds.
Though really, even with the stock cooler, the system shouldn't be set to ramp the fan up to 100% at 50 degrees, because that's completely unnecessary, and isn't going to keep these maximum temperatures down any better than if it were set to reach 100% at 80 degrees. Setting the fan curve up that way is just going to make the fans ramp up and make excessive noise even under light loads that don't require those kinds of fan speeds.