I think this is where your logic goes wrong:

$(C_t − P_t − S_t)e^{r(T−t)} + K$

With reference to the above equation, you are saying that "...To that money that we owe, we add the money that we owe to the contract buyer.."

Yes, $(C_t − P_t − S_t)e^{
r(T−t)}$ is the money that we owe, but $K$ is not referring to money that we _also_ owe the contract buyer. $K$ is the strike price, so it is the money we **receive** from the contract buyer at maturity when he exercises the option. Yes, the call option is in a losing position, but $K$ is not referring to the actual loss.

So, the equation is really saying that the money we borrowed to finance this strategy (plus the accrued interest) is less than amount we ended up receiving when the contract buyer exercised his option and bought the stock from us at the strike price. A net gain will also result if $S_t < K$ as well, which demonstrates that an arbitrage opportunity exists when $C_t - P_t > S_t - Ke^{-r(T-t)}$.