How important is the quality of power cords in a high end system?

A year ago I was looking to buy a secondhand amplifier and found what I
wanted at a price I was willing to pay but it had an "improved" power
cable. I had two problems: firstly, people who do this sort of thing
are unlikely to be competent at working with electronic components and,
secondly, if someone saw me using an amplifier with a fancy power cable
it would be embarrassing. Replacing power cables devalues the resale
price of an amplifer (as do most modifications).

Does it really make a difference in sound?

What are the chances that a company cabable of designing and
constructing an audio amplifier is going to use an inadequate power
cable?

It does raise an interesting point though. When the nonsense around
speaker cables took off, speaker companies had to stop supplying
speaker cables. If they supplied normal wire it put off audiophiles and
if they supplied fancy cables the more technically competent would be
put off. Have the manufacturers of audiophile amplifiers started
supplying amplifiers without power cables?