Do good power cords have to be stiff?

Any power cable that is 12 gauge is sufficient. Stiffness does not matter. The metal in the Romex that supplies your outlet is either 12 or 14 gauge solid copper in the USA.

A cable made of braided copper wire with a gauge that is equal to or thicker than your main’s cable is sufficient.

Depending on the equipment you are powering and it’s current draw …you might even be able to get away with 16 or 18 gauge.

@drbond Until very recently I would have agreed with your assessment of power cords making a difference.  Out of morbid curiosity I picked up some very inexpensive (fairly stiff to address the OP) W Audio cables on Amazon.  I have three of them in my system now, but will only say with conviction that it changed the sound of my Pathos Classic One MkIII amplifier (with 1960s Mullard tubes).  I noticed the change immediately and can't imagine that any audiophile wouldn't able to also hear the difference in my system.  Based on that experience, I have added Audioquest Chicago interconnects  (replaced BJC LC-1) which also made a clear difference to my ears and was also immediately noticeable to me.  Based on that experience, I added an Audioquest Cinnamon digital coax and I can't say that I could tell any difference between it and the BJC silver plated cable that it replaced in my system.

Verastarr flat foil cords are incredible sounding and very flexable.

+1 on Triode wire labs.

Or, Dynamic designs - some of the iterations are stiff though....there's a digital Challenger listed that is fabulous (and flexible...).

i suspect that the Cables zavato is using with male ends will tend to be more on the stiff side 😂

‘Sorry.  I couldn’t resist.  We had an ice storm in the northeast and I’m suffering from cabin fever.