I think this might be part of the problem. The idea of handing over your private messages to some random company is so absurd that the average person probably assumes that it couldn’t possibly be what they’re asking for. You’ll have to make the permission prompt really explicit to overcome that.

Plenty of people installed Facebook apps that required permission to read all text messages

Yes, but that doesn't mean that they consent to Facebook using that data. Whatsapp for example asks "read SMS" permission to verify your phone number. It also requires access to your contact list or else everyone's name will be shown as a phone number.

The average user seems to assume that the information is used to provide the service and nothing more. The last person I asked if they were worried about how much information FB has about them replied something among the lines of "Yes, but it's only things that I deliberately chose to share." (paraphrased)