First part of your theory is true. The natural defenses do generate heat via inflammation. Which usually does nothing good or bad, unless it sometimes gets really bad.

Beneficial part, mostly no. You aren't cooking virus or bacteria, and heat doesn't make the defenses work faster.

However, the really interesting fact of the day is body temperature does kill fungal infections. >36.6°C will kill fungal infections. Majority of fungal infections are on the outside of your body where it's relatively cool, but if they get into your blood, travel up inside your urinary tract or reach some organs, they are incredibly difficult to treat.

Average human body temperature is dropping over time. Humans were constantly sick with some persistant virus or bacteria, an average body temperature was about 37°C (above the fungal limit). Then modern medicine started to fix those persistant infections, and average human body temps dropped to about 36°C (below the fungal limit). As a result, in modern times we find a lot more internal fungal infections that are difficult to treat.