Am I missing something here, or are they really terrible? Since I often play with random civs, my response to getting Byzantium in the future is likely to be "Exit to Main Menu".
I've retired from 2 Immortal games as Byzantium recently, but I thought they went wrong due to my noobiness and micromismanagement. I don't want to say terrible, but you really have to get lucky with religion. And despite me re-playing those games over and over, I never got Sacred Sites because it was first-second reformation to go. Having Ethiopia/Celta in your games pretty much makes you want to bang your head against the wall. I will want to re-try Byzantium and win a CV with them for once.
I've read (above) that you just lose your reformation belief if you take that pick too early. So I went with gold for temples, which did nothing for me in the short term. That is the downside of Piety. Several picks are going to be useless at the time whereas everything in Liberty provides an immediate boost. The same is largely true with Tradition. Ideally I'd like to defer that pick until I had a temple almost up.
Temple Policy does nothing early game, so I find it better delayed. If anything, Tradition opener/legalism/aristrocracy are all good fillers. In fact, I tried opening Tradition, then Piety for shrine building, then all the way to Reformation, picking mandate of haven, organized religion, and religious tolerance, finish Tradition and then finish Piety. But I am not sure if that was optimal. For all I can say, finishing Piety with the temple policy is the best choice, since it will be late enough, and you can choose to plant that great prophet to get the gold generation going right away.
It is also possible to save policies, if you tweak pre-game options, but saving policies in a CV game is dubious. Also, Rationalism is important, so it can be a"filler" for a while, depending on how fast you tech/culture. No?
