Average "healthy" body temperature is down more than half degree exactly because of reduction in inflammation. People used to have very unsanitary lives and were fighting infections nonstop. 37.0C was the "normal" body temperature when the "norm" was first discovered in the early XIX century, now it's considered to be a sickness already.

I'm not an expert but maybe keeping the immuno-army busy with constant infections keeps it from going bored and attacking itself

also not-an-expert opinion, but an average higher body temperature would reduce immune system workload by virtue of killing a lot of stuff dead via inhospitable environment before an immune reaction could ever muster..

Fascinating claim, where did you hear this?