Most live publicly with their faces on display for all to see and others taking it a step further, participating in Facebook alongside billions of others.
It doesn’t scream facial identity being a major concern.
Most people have them in their homes, breathe them, eat them, and others take it a step further, participating in the creation of them.
It doesn't scream fear of cancer being a major concern.
That people live their lives accepting that their faces are on display is not evidence otherwise, since there is literally no other option.
Participating Facebook is also not evidence otherwise -- at most, it's evidence that people are willing to trade privacy in some circumstances (and I think even that's a bit of a stretch), but I'll bet that most Facebook users would object to having their privacy invaded without their consent -- which means they care about privacy.
I say this with seriousness. When considering this alternative, the option of living alone, without human interaction, public identity shows it positive attributes.
Which is what was being proposed and subsequently doubted: that people were willing to consensually trade their picture for $5.
Questionnaires where they reveal things, often associated with at least a way to contact them for an interview, but sometimes with name and everything, usually is psychiatric settings (where people are often incarcerated without any proof, trial or any of that I might add). Things like whether they stole from their employer. Whether they ever used violence to obtain sex. They often ask children, homeless, prisoners, patients ... other groups with perceived or real precarious situations. (things that would never pass an ethical review board for, say, medicine)
https://onlinelibrary.wiley.com/doi/abs/10.5694/j.1326-5377....
https://ajph.aphapublications.org/doi/full/10.2105/AJPH.91.1...
https://psycnet.apa.org/record/1991-07467-001
https://link.springer.com/article/10.1023/A:1020007004436
So yes, I would say that a lot of people are willing to give up a LOT more privacy than a face picture for a small reward.
But the fact that that's true doesn't mean that people don't care about privacy.
You could even argue that it's indicative that people do care about privacy, as they attach a material value to it. This isn't an argument that I'm really making, but it isn't an unreasonable one.