You are misunderstanding. With religion being the big exception most people believe in things they have found support for. Am I “inherently close minded” not to believe in the existence of Santa Claus or Zeus? I could be wrong but since I have seen no rational reason for believing in them I don’t.To believe the absolute nonexistence of something is inherently close minded. Not open minded
The "internally inconsistent claims" you say exist are often, actually very consistent among the educated practicers of their faith. Many things are not straightforward. Not everyone is a theology major in their own religion (as great as that would be) so the "gotcha" moments on the average layperson is not evidence of inconsistency. It's evidence of a lack of education, which absolutely exists in every collection of people on every topic.
To your point on inconsistent claims though, some of those are in fact the result of the large amount of denominations that exist.
And some things that are not settled are simply deemed "mysteries of faith". I can see how those can be construed as an "inconsistency" from the outside looking in
Atheism is simply a lack of belief in gods not a claim that gods are impossible. I would welcome being proved wrong.
And yes inconsistencies and using the mystery of faith as a catch all to cover for those inconsistencies moves the needle more towards the nonexistence of God.