When is the last time you had to give your PNR to a check-in agent? There is absolutely a way to fly without it and it is also true that it is a privacy risk to transmit it since if someone can get hold of your name and PNR, they can find a lot about your trip, maybe even sabotage it.

Your e-ticket has been issued, anyway the original one definitely was. Call a contact number of a smaller country during off-peak hours to retreive your PNR or use FFP number (if entered during booking) to locate.