Everything I have ever read said it does not matter whether the oil is synthetic or not. For a flat tappet engine you need the ZDDP. Every cam manufacturer I have seen a statement from agrees.
Not so black and white. The comments I read - including the one posted above (albeit it being a bit outdated, apparently) - appear to suggest the necessity of the zinc only during camshaft break in. Whether the engine needs it afterward seems to me to still be a point of contention. I'd personally rather err on the side of the zinc if it does not mean sacrificing something else.
And from all that I have read thus far, the best solutions on this side of the issue are:
1. Valv VR1;
2. AMSOIL.
Whether STP has enough in it is still up for debate.
If I can, I personally would rather go with AMOIL. I say "if I can" ... I certainly can, but what I mean is, if it is a good enough oil. Some here like it, some do not. I prefer synth oil to non-synth - my personal experience with synthetic before this lack of zinc issue came about has been very good.