Any wire with rubber insulation which can move or bend under normal use should be replaced. Especially where two wires are in close proximity to one another. It's not so important with single wires, say under the motor board where movement is minute and very limited, but things like a mains lead I think is a must.
Even if the insulation appears to be sound, you really cant be sure that somewhere along a 1 or 2 metre length of wire, the insulation hasn't had something dropped on it or spilt on it in the last 80 or 90 yrs, that will make the rubber deteriorate quicker than the rest, and in 5 yrs, or 5 months, you might end up with exposed wires.
I dont think anyone could honestly say that replacing a power cord with a modern reproduction would in any way distract from the look or value of any antique electrical item. You can always wind the original wire around a toilet roll tube & keep it in the bottom of the case, and reconnect it later if you want to sell the machine.(if you were that worried about originality or value)
If I was looking at 2 identical machines for the same price, and one had good original wiring & the other had authentic replacement wiring, I know which one I'd be taking home

Infact, personally I'd be willing to pay a little more for the machine with new safe wiring!