One reason that occurs to me is that we have examples of flint chips that aren't Clovis points, so we know what a non-designed chip looks like. As far as I know, we don't have any examples of flagella that don't work right or have parts missing or whatever it would take to decide they weren't designed. We do, as I understand it, have examples of the parts of the flagellum being used for other things, but IDers reject that as evidence that the flagellum could have developed from those parts.
I suppose one could argue that we don't know for sure those non-Clovis chips aren't designed, too. But the identification of Clovis points as designed items isn't just based on what they look like--it's confirmed by finding them in association with other evidence of human habitation and/or with the remains of animals they were presumably used to kill and dress. Someone who wanted to assert those other chips might be designed would have to bring some evidence beyond a simple suggestion.
Unlikely, impossible? How do we decide?
Well, you could try it and see if what you had to do to get that "natural" shard matched any known processes. How much heat will fire clay and also shatter a stone? Does a forest fire generate that kind of heat? Is there any other evidence of a fire? Is the surface of the fired clay smooth or show fingerprints, or is it rough, the way the boundary between dried clay and undried clay might be? Are there lots of examples of this kind of shard around, indicating that it was a common process, or is this the only one, indicating that it might be an accident? Is it found in conjunction with other signs of human activity?
I guess in conclusion I would say that the things we seem to produce with such ease, points and pottery, are really more difficult to produce than we realize.
They only seem easy because there are so many bits of information that we have and take for granted.