Actually, you have the order of events a bit wrong. Interferons are the among first response to infection, along with production of inflammatory signals. In the case of a new virus, like SARS-CoV-2, there are no memory B cells to be reactivated and make antibodies. So, antibodies come later in the infection process. Indeed, some infections can be cleared by the innate immune response, which includes interferon signaling, without even stimulating a B cell response. Type I interferons (like IFN-alpha) are generated not only be dendritic cells but also by infected cells. Indeed, producing interferons is one of the first thing that infected cells do. The interferons then signal to the immune cells to start the immune response, which includes production of interferons by the immune cells.