Is it actually unsafe to travel by plane in the 3rd trimester?

No: Air travel is not risky for mother or her fetus. However, restrictions are in place to limit the possibility of an unplanned delivery on the plane. Check with your airline regarding the rules. Typically domestic flights <36 weeks; international < 32-35 weeks. You will need documentation from your doctor about your due date. Ask for a wand/pat down to avoid accidental radiation at tsa screening.