Aww, sad :(. The faces are likely problematic, but the voices themselves shouldn't be (not identifiable to a person)

My wife's in a remote master's program. What they did is chop up the lecture. When students asked a question, that was omitted from the online presentation. The result is that an hour of class winds up being, say, four 12-minute videos.

That sounds weird, and it can be a bit choppy, but it has advantages. You get to stop and think at the points where others needed to ask questions. You have natural breaking points if you prefer smaller-sized bites.

So, I don't know if the students are in all of your video, or just when they ask questions. But if it's only when they ask questions, you might consider this approach.