This description does not discuss reliability. IIRC, these cat's whisker diodes need frequent readjustment. (http://en.wikipedia.org/wiki/Crystal_radio#Crystal_detector agrees with that)

Using a single one to build a primitive radio receiver is easily doable, but combining a couple of thousands of them to get a 4004? I doubt one could ever get that to run for a second.

So, they would have to step up to better diodes. Of course, one can speculate that they could have done that, and they could. In the end, it is all down to the observation that flattery (aka imitation) is way easier than invention.