This is only true from the Protestant Sola Scriptura perspective. The Catholic and Orthodox traditions hold that God still directs the Church through the Holy Spirit, which makes the documents that you identify only some elements of that direction—an output of the authority granted to the Church—not its final form. So, no, it's not accurate to say that we are far removed from any claims of direct involvement from Deity—several branches of Christianity hold that He is still actively involved.
> If the only meaning someone sees in religious guidance is its connection to a literal physical deity, they're in for an exercise in frustration from what is at best the world's longest game of telephone.
Again, it's not a long game of telephone if God is actually still directing the Church today. If you accept that He guides leadership right now through the Pope and the Bishops, which is the stance articulated by OP, you're at most a few steps removed from His regular guidance.
Which gets back to my original point, which is that this really all comes down to which axioms you want to accept. All religion is unfalsifiable, as you observe, but falsifiability cuts both ways and you can't logic your way out of that to logically conclude the absence or irrelevance of a God. What you can do is decide which axioms you're going to start from and work from there.