I do not follow on the first page this boldface part of the sentence:
By the foregoing reasoning it should be clear that we are only allowed to reason with known true statements, that is, we are not allowed to assume that a proof for $\varphi$ or $\psi$ exists. Doing so, leads immediately to a wrong conclusion.
I'd say that if a proof exists then $\psi$ is true ? So what is the distinction between provable and true, does one imply the other aka Post theorem ? Why this doesn't work in game theory ?
I am unsure of what exactly you are trying to ask but what I can say with regards to the question is the following:
You are correct in that if a proof exists, then the statement is true. However, the text is saying that the converse is not necessarily true. In other words, a statement being true does not require the existence of a proof. Take for example, the existence of 0 such that 1+0=1. This is true, but there is no proof for it, because we defined it this way axiomatically. Due to incompleteness, there will always be statements in any logical system that are true and cannot be proven per se.