Can the innate immune system defeat a pathogen by itself? Fast enough that we don't develop antibodies, etc.?
Generally antibodies are formed, but it is possible that innate immune system has cleared out the pathogen. But antibodies will be formed even after that, as adaptive immune system has received  the stimulating signals from helper T cells, which are matured during the innate immune responses triggered by an infection. 
This signal triggers B cells to produce antibodies. Antibody production is activated because of signals from innate immune system.

So, even if infection is cleared by innate immune system, still antibodies will be synthesized.

Reference:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3399021/

https://medicalxpress.com/news/2020-05-dont-antibodies-immunity.html