I haven't heard the latest about periods, but I want to point out a potential fallacy here. It's said very often that the easy proof of the existence of transcendental numbers (on cardinality grounds) is non-constructive. But, that's false! It is constructive. Given pen, paper and lots of time, I could extract from that argument the decimal expansion $0.a_1 a_2 \ldots$ of the transcendental number that the proof constructs.
See, for example, these comments of Joel David Hamkins.
I suspect that the same is true for periods: that there's an effective enumeration of them, so there's an algorithm for generating the decimal digits of a number that isn't a period.