Not exactly. Briar uses bluetooth or wifi when peers are close, but also tor (over standard internet) when not, so it's possible to use it at wide scale.
I would think that a meshnet only over contacts would have too many holes to really be helpful in the hypothetical protest setting that's being described. Definitely a cool way to do it! Do you happen to know if it's XMPP or something, or it's own protocol?
There are still holes of course, connecting only to contacts limits the spread of messages but ensures you don't leak too much information if your device is compromised
Briar actually does set up a meshnet for groups and forums, so long as people are contacts of each other. See the diagram here: https://briarproject.org/img/howitworks3(mobile).svg