Should power cords have filters on them?

Best world scenario is if your system doesn't need filters, that's all. Use it if you need it.

Fine caps across the hot and neutral can help, without veiling or dynamic limitations. I use good ones in my PoBoxes, for example, but not on PCs per se.

I have tried ferrite clamps and it did help on my preamp. Still think a good power conditioner is better.

Although power accessories, inline or outline, such as power conditioner, surge protector / suppressors, ferrites, choke, parallel filters do help cleaning the power that is fed to your equipment which do help them perform better; a good enough equipment will have a good enough power filteration circuit within their power supply, therefore, minimizing reaction when changing to a very good power cords. In theory, the better equipment it is, the less a good power cord will react it's performance.

For your discussion about whether if a power cord should have a inline device for filteration... My thinking is... a good power cord design should focus on how to prevent interferences instead of how to filter them. Good manufacturers who makes equipments should take care of the filter problem.

Power cord filters are required (by law) in some jusisdictions (Europe, as usual, is ahead of the USA) to prevent equipment from putting noise onto the power lines. This is the opposite to what audiophiles are worried about. Such a "filter" is properly called a "retlif" ("filter" spelled backwards".