The issue wasn't consistency with what we see, but self-consistency of the concept of "all-powerful."
But it is still us, humans, who judge whether a concept is self-consistent or not.
The matter isn't resolved "objectively, on its own".
Unless, of course, we are fully sure our powers of reasoning are completely accurate and in line with "how things really are". Which is rather arrogant, IMO.
That God can't create a stone so heavy that God can't lift it isn't a practical problem unless God actually attempts that.
How do you know God hasn't done this?
But it is a practical problem for people who want to include both "all powerful" and "logically consistent" in the definition of God. You have to pick one or the other, or you're liable to end up like LG.
Inasmuch can a criteriium like "logically consistent" be applied to a person?
Can humans be considered "logically consistent"?
Mathematical systems can be "logically consistent". But is God, or are humans, mathematical systems?
If I wanted to actually prove that God doesn't exist, based on what we see in the world,
That's allright, but I think you are including a very limited scope of information here.
We see all kinds of things in the world, and if we are to make any judgments about God, based on this world,
then we have to include everything we can see (hear, smell, taste, touch, think of) in this world. Which is a lot of things, including all religious scriptures, buildings, activities etc. for example.