I think there’s some vagueness here about what “big brains” means, and what “tool use” means.
Chimps, Gorillas, and Orangutans all use simple tools. All have very big brains. But their brains are only about a third the size of modern human brains.
Is the OP complaining that you need bigger brains than Chimpanzee’s to use tools, but you can’t evolve a bigger brain that a Chimapnzee’s without the selective pressure of successful tool use?
But that doesn’t make sense. Chimps and humans use their brains for many other things than tool use, just like most animals use their brains for other things than tool use. Is the OP expecting a fossil hominid with a chimp-like brain that used stone tools, and since such a fossil doesn’t exist that therefore evolution must be false? Or is the OP expecting a fossil hominid with a human-like brain that didn’t use stone tools, and since such a fossil doesn’t exist that therefore evolution must be false?
There does seem to be a pretty strong correlation between brain size and tool use, from smaller-brained chimps that use tools only occasionally to Homo habilis with a larger brain that used crude tools, to Homo erecuts with an even larger brain that used tools consistently, to Homo neanderthalenis with a brain the size of modern humans that used sophisticated but uninventive tools, to Homo sapiens sapiens who can create elaborate and inventive tools.
I’m not sure how that proves evolution false though.