"Lead with Culture" is of course a major theme of ACTFL's current standards. The way I interpret this is that the end goal of the lesson needs to be understanding something about the Roman world and that the language is always just a tool to achieve that goal. Without cultural knowledge, a student can't really understand many texts, but more importantly, interacting with the culture is the reward for learning the language.
The difference for the Latinist is really just how deeply foreign Roman culture is and what a profound exercise of imagination it is to explore it without living representatives. I encourage you dive into the religion, the magic, the graffiti, the toilet humor and don't just stop with the poetry and histories. Part of the advantage of this is that it is fascinating and often more relatable for many students, but also the related texts are easier and more accessible. See my novice authentic texts database for some places to start.