This could easily be added by having a privacy / visibility selector. So a reply can be left at default (public) or changed to "private" (i.e., DM). Any further reply underneath is then private / DM (only).

This is pretty much how Mastodon handles DMs and it works pretty well. If there's no post or comment to reply to someone, then a reply on their profile (if they have one) would work to initiate a DM.

An alternate approach to this would be for SN to simply accommodate the comment (an optional field) on Lightning Address payments and display that comment in /notification when displaying that payment's "sats were deposited" notification. So for me to DM you I would send from my LN client to the Lightning address for your SN account and that would let me send my direct message to you through that. You would see that there is a new notification and then from that you could view my message in your notifications. You wouldn't know who it was from unless I gave you my SN nym, or Telegram, or whatever. If I added my SN nym, you could then reply by sending to my Lightning address. It's not threaded messaging, but this is minimalist approach requiring very little modification on the SN end.