Many observers do use iNat to document behavior. As @pisum notes, some behaviors are easier to document in video rather than with still photos, but often it is possible to document things like interactions between individuals (hunting, courtship, fighting etc.) using a series of photos.
It is a bit more complicated to systematically track different types of behavior because this is not an integral part of an iNat record. The best way to collect observations of a particular behavior are by using observation fields or projects. If you have observed an animal with a behavior that you think is interesting, it may be worthwhile searching these to see if someone has created a relevant project – often this is the case. The thread on favorite projects should give you an idea of the spectrum of themes covered by projects. You can also create your own project if you can’t find one that is suitable.
By the way, it is perfectly fine to repeatedly observe common organisms if you find this meaningful.