Are you sure? I expect they treated syphilis through malariotherapy, not influenza.

See the following papers:

https://www.nejm.org/doi/pdf/10.1056/NEJM199004263221713

https://jamanetwork.com/journals/jama/article-abstract/398731

I have a hunch that fever therapy could be useful for many other infectious processes, autoimmune disease, and cancer based on this history, and probably safer than administration of a live organism like malaria (even if it is the less pathogenic species), too.

Fever therapy was widely used in the US before WWII and the advent of antibiotics. It was used for all sorts of things and a variety of bacterial vaccines were deployed.