So many people don't understand that a pyrexia is a self-defense mechanism for the body. It's reflective of the absolute failure of our public education system, especially in the area of science education.
If Aspirin was just discovered, it would never be approved for over-the-counter use. As you point out, it can have dangerous and even fatal side-effects.
But aspirin is a wonder drug. Judicious use of it can be very effective. In my previous post I spoke of the over the hump phase of a viral infection. That is after the fever is broken and the chills have stopped. At that point the viral replication should be over and a little aspirin for pain relief should be OK. It wouldn't be my choice.
But prior to that you are right about its negative effects and it also suppresses the immune system. Any relief of discomfort you get would be at the cost of a longer and stronger infection.