We are already doing this, no?
Every time we take medication that lowers our fever during an infection (antipyretic ), we trade a bit of immune response for a much more comfortable illness.
The other aspect of this is that carrying more pathogen, for longer, increases the chance that you will infect others.
Or that it could do more damage, depending on the type of infection. For example, HPV os though to only lead to cancer for those who have a persistent infection for a few years.
My personal theory is that fever arose as a specific mechanism against specific pathogen that humanity encountered some time in the past. Probably very lethal. The only people who survived were ones that were getting fever when encountering large number of various antigens. That pathogen I think is long gone and most human pathogens are completely not influenced by the fever. But the mechanism largely remained.
Or perhaps fever works against bacteria so if you keep yourself isolated when you are having viral disease it's useless, but if you still contact people when virally ill maybe fever protects you from secondary bacterial infection to some degree.
Fever, I suspect, is the body trying to exceed the operating temperature range of the pathogen to either kill or impede it.
I don't know if there has been any research on this that either confirms my genius theory or disproves my rambling uninformed delusion.
The body also does not go into fever mode for every kind of infection so I personally think that it's quite adapted to only "fevering" for the right kinds of pathogens, as in: yes it does still help/do something but there can of course be "imposters" that make you go into fever mode without it helping much.
Some people get fever that goes way too high and kills the host if we didn't have Ibuprofen and the like (or herbal remedies to the same effect). If none of those were available, the genes affecting this would naturally die out/become less prevalent. Which is along the lines of the article that "letting diseases run their course" might be beneficial in that sense.
There's no evidence of this tradeoff.
You cause damage to your body through inflammation. You invoke deleterious pathways that normally sit idle.
Getting sick primes the immune system for defense, but it is also damaging to healthspan.
The Hayflick limit really isn't relevant here.