There would need to be a way to guarantee that only the browser could do this, or at least some way to tell exactly what the source was.
I don’t think there’s a way to do that. Let’s say browsers implemented this. I could then just take a copy of Firefox source code and make my own version, which is exactly the same than normal FF except the fancy screenshot tool has been slightly modified to allow editing the page before taking the screenshot.
In addition to what others have already pointed out, please also note that mentioning any other account in a “private” message chain will allow that account to retroactively see all the messages in the chain.
Basically DMs in Mastodon are pretty useless.