The cheap minimalist SS/fixed bikes are built around low-end frames. That's not to say such a frame will break or won't ride well, because plenty of miles ridden on very low end machines tell me a cheap frame can ride well, and also take a lot of rough treatment.
Where they fall down is weight. One of my leftovers from a donor is a 21" Apollo welded hi-tensile 700c hybrid frame. Stripped of everything apart from the BB and headset, it weighs 9.6 lbs. Compare that to my 23 1/2" Ian May tourer frame, in the heaviest ST variant of 531, which weighs 7.8 lbs with BB and headset. My other 531 frames will be marginally less than that, so the difference between a cheap plain gauge hi-tensile frame and a quality double butted one is going to be around two pounds or more - which is roughly the weight of the gears you haven't got on a SS. So put it like this, a cheap SS will weigh roughly the same as a quality steel ten speed, but be much less versatile!
If you just want a very simple single speed beater to ride a couple of miles to the pub on without doing much maintenance, the heavy frame might not matter, but to me the whole appeal of SS is not only mechanical simplicity, but also lightness - and to achieve that you have to start out with a nice light steel frame, preferably a lugged one with a 531 sticker on it.