Should Daniel and I live-stream our wedding ceremony on social media?

Yes

I think it depends on the reason for wanting a private ceremony. Is it because you only want to share it with you closest loved ones, or is it because you want it to feel like it is only being shared with your closest loved ones. Based on what I perceive, I would opt to say that you should not live stream it, though I know there are many who would want you to. (Although I'm going to be there in person, so I am also slightly biased)

No