Let’s start with your first three statements:
1. If a God exists who is omnipotent (all powerful) and omnibenevolent (all loving),
2. and evil exists,
3. then God cannot be omnipotent or omnibenevolent.
The atheist implies with this argument that an omnipotent (I like your definition and that is how I’ll be using it) God must cause all evil to cease. Since we don’t have to look very far to see evil in the world, then God must not be omnipotent and omnibenevolent. There are some problems with this argument. What exactly can be considered “evil”? Why must God stop “all” evil?
When I took my 6 year old daughter to the dentist, she needed to get a tooth pulled out. And from her perspective, I let the dentist stick a painful needle in her gums. Was that evil of me since I permitted her to experience pain? Why didn’t I stop it? From her perspective, it might not have made sense unless she understood the purpose in the pain.
So, sometimes there is a greater good out of what we might construe as “evil”. And this is consistent with the Biblical description of God.
I think the atheist argument would stand if the following statement was true.
4. God has done nothing to address evil in the world.
If God is omnipotent and omnibenevolent and evil exists in the world and God has done nothing to address the evil in the world, then it could be concluded that either God is not able to address evil or God does not care to address evil.
And, of course, point 4 is easily refutable.
As for defining omnipotence as the ability to do anything including doing illogical things I believe is a straw man argument. If one accepts this definition, then of course this type of God does not exist. Nor is this type of God described in the Bible. And it is not a God that any theist is trying to show exists.