It does, but GM-CSF is released to create more phagocytic cells because they are being depleted by the infection - DCs definitely die off due to increased apoptosis in sepsis (this contributes to post sepsis immunosuppression) and in severe infections like severe influenza. I suppose this is far more relevant for something relatively severe, with heavy symptoms that you take days to fight. An acute infection with little symptoms is not likely to do this, but I'm still dubious about saying you have more effective innate immunity during an acute infection given how common confections are too... Active inflammation also inhibits other immune functions like for formation of germinal centres, some t cell functions etc, so having extra neutrophils might help with some aspects but not others.

An example is BCG vaccination (a live vaccine) induces emergency granulopoiesis which confers short term protection against bacterial sepsis, so an infectious 'nudge' like thia to the immune system can help it in the short term but you don't want to whack it with a baseball bat (eg. Severe influenza).