But there are real-world instances that contradict this hypothesis already. Humans don't exhibit Turing-computable behaviour because they did indeed solve problems that aren't solvable by Turing Machines.
I can give a very concrete example as well. Consider the following program (in Python for simplicity's sake):
def nat(x):
yield x
yield from nat(x+1)
for a in nat(2):
for b in nat(2):
for c in nat(2):
if a*a*a + b*b*b == c*c*c:
print(f"{a}^3 * {b}^3 = {c}^3")
exit()
Yet it has has been proven centuries ago that the above program won't halt.
The mathematicians couldn't have used a Turing Machine-compliant computation, so something else must be involved.
Now this of course doesn't mean that humans are generally capable of solving every problem that's not solvable by an algorithm, but it means that they can solve problems that algorithms can't solve.
Yes, there is. There's a machine which does nothing but print that exact proof.
You seem to have conflated solving an instance of a problem with solving the entire problem. Turing's theorem states that no one Turing machine will get the answer right for every instance of the halting problem, but for any instance that one specific machine gets wrong, there's another machine that also gets that instance right (in addition to everything else the original machine gets right).