Absolutely no experience, so this may be stupid.
Has anybody tried to replace this with a "daily shared morning tea/coffee"? Create a collegial atmosphere that's intrinsically unsuited to active management; there's no real need to keep people ontopic anyways because they'll naturally talk about work. Plus drinks may keep people occupied enough that only one person would naturally talk.
[edit] Oh, is that what the watercooler does!
It's a lot easier to be collegiate in a college.
One of the main goals of Agile processes is to remove overhead from the developer so they can focus on adding value to the stakeholder. Anybody who starts goofing around the standup is disrespecting and wasting the team's time.
Now, your idea might be worth a try. As I said in another post: the team runs itself and might decide to try something like you propose... and keep doing it if it works for them.
But by all means try it out; if it works for your team, great!
You're right; a better phrasing would be that the manager is not an active participant in the meeting. It's a developer coordination meeting, not a status reporting meeting. So the manager is welcome to listen in, but they're not supposed to be managing during that meeting, and in particular they should not be asking questions, because that'll very quickly turn the standup into a daily status report instead.