Even FPSes are implementing achievement systems that do not affect game play, or minimally affect it. For instance, in Half-Life 2: Episode 2 you get an achievement for carrying a garden gnome around for a long time and eventually launching it off in a rocket. Does this need to be done? No. Does it give you anything other than an "achievement achieved" message and the right to brag about doing it? No. Do people still do it? Absolutely.
Let me give another example. In Team Fortress 2 you can earn achievements and you even get other weapons for getting more achievements, but these weapons are not necessarily better than the weapons they replace. (Eg, You get a new weapon that slows someone on hit, but it does less damage). Without getting into whether or not all the weapons are equal in power to the weapons they replace, that is the goal of the system.
Addressing your World of Warcraft example, yes, a lot of rewards are based upon time over skill and DO affect gameplay, but not all do. There are many eye-candy, prestige rewards, such as rare mounts (that are no better than other mounts), vanity pets, tabards, titles, etc. Some people take to trying to collect all of the vanity pets in the game, and while it has no affect on gameplay, they do get some sense of accomplishment out of it.
While I agree some systems reward people for doing non-skilled, repetitive tasks, I do not agree with you when you say...
[a loot system] would have to be a system that doesn't affect overall game play, and as yet I have not seen such a system in play.
That isn't to say you didn't have some very viable points, and I do also acknowledge that you likely were referring to whole-loot systems, instead of sub-systems, but I feel that many of my points still stand.
