The heat protectors are there on the wires, I can see them in your pic. They are not overly large but they are there. Just gently remove them and work them back over the new cord.
I have several Singer foot controllers that do not have the heat sheaths and so far they haven't melted the wire. Modern electric cord insulation isn't made from the same material as the old stuff was either so it may not be as sensitive to heat.
Joe