"That" is not an appropriate word in this sentence.
Yes it is.
We can't use "that" in the place of "who".
See my previous post above. Using 'that' in reference to humans is not an error. It is a question of style, not grammar. While most editors prefer 'who' for people, there is no rule saying we can’t use 'that', which has been widely used in reference to people for many centuries. It remains so today, especially in British writing.
Labour insiders argue that Balls is probably the man that
can most effectively deliver a reshaping of the party’s economic policy. [The Guardian]
This demonstrates how completely out of touch they have become with the people that
they purport to speak on behalf of. [The Daily Telegraph]