The chain is directional yes, but it looks to be fitted correctly. The writing is supposed to be on the outside. At least that's the way with the Ultegra chain.

So the sprocket is machined out of one billet... No possibility of mismatched cogs then. One would assume that they know where to put the gates and the pickup ridges.

Maybe a SRAM chain would work better. The Shimano ones are quite special.

Take the chain off and eyeball the derailer, that it aligns exactly with the intended cog in each gear. (I spent many hours just this weekend fixing bad shifting; somehow the supposedly matched XT shifter over-reached the derailer such that alignment was not perfect over all cogs. I modded the derailer quite a bit to fix it.)