The reason they didn't fly to Athens is because most of the places they visited are not in Athens itself.
In the folloing map you can see that. #1 is Patra where the boat arrived from Bari. #2 is Arachova where the detour took place. That's about halfway to Athens. Then #3 is the Corinth Canal where they did the bungee Jumping (RB?) and # 4 is Athens-pitstop.
There are two things I'm more interested to find out. First how did they get from Venice to Bari? They are a long way from each other and they would need to take either a plane or a long train ride. And why didn't they take the direct boat from Venice to Patra even if takes 24 hours to arrive?
Second what were they doing at the Panathenaic Stadium (where Redmont falls )? Is that a RI before the pitstop and what's the point of that? Because I refuse to believe that it's the RI before the RB, it would be insane to face the traffic in Athens and then leave for the Corinth Canal only to come back for the pitstop.