From that point of view, there should actually be a connection to the vent.
Yes, but the vent works as an entire system. If, for example, negative pressure occurs somewhere (also e.g. low-pressure outside air), that can have an impact on it. You need to have the respective air volumes checked to see if they are properly designed. If it were a "normal" floor drain, I would say you could just pour water in there because it might have dried out. You alone wouldn’t be able to manage that.