I agree synthetic statements are definitive. They are also tautologous if they are truly synthetic statements, including all of mathematics. And I agree no analystic statement is immune to revision.
Personally I don't find anything I can say about the subject of God that I would call well founded.
Provisionally I would say that existence is a mystery that seems to require some sort of explanation. And the idea that consciousness arose from nothing and has no purpose beyond the purpose it makes itself, seems like a leap of faith in some sense.
I like Rowan Williams' comment in the last video where he describes the encounter with God to a footprint in the forest. From just the footprint, you know it's big, you know it was there, it leaves an impression, but beyond that there's not much you can say for sure about it.
And it's interesting that none of the arguments "new" atheists use are anything new. The Bible itself already complains that God stands by and allows suffering. It already asks why. It already says that such a God is incomprehensible and it seems wicked. None of these observations are new. Believers incorporate these difficult problems into their questions about how God is, whereas atheists use them as a basis for saying God cannot exist or he is evil if he does exist. Sometimes it's not clear which.