If you have the 15-30 minutes to spare after your flights, I'd just check them in. No stress, and you won't have to schlep a (from the sound of it) extremely heavy bag around the airport!

The documenting of bag contents is just for some piece of mind if you are nervous about damage or lost items. Me personally I would worry be and would check my luggage in a heartbeat on such a simple itinerary. A bag getting lost permanently is extremely rare. And in hundreds of flights to and from Greece that has never happened to me.
I've had my bag not arrive in Athens flying in from Paris two or three times - but that was due to either an extremely short connection in Paris (delay on inbound flight) or strikes (which they love doing in France). And in those few cases that my bag did not arrive with my flight, it was delivered to me (about 350KM away from the airport!) the next day.

Oh and if you want some extra piece of mind and happen to be an iPhone user - throw one of these AirTags in your bag and you should be able to have a pretty good idea about what's going on at all times.