After seeing that FB listing I can see why $600 is making more sense.

Seller doesn't know the exact year or size, so probably not the original owner. Bike has disgusting bar tape, mismatched components (including a very old and beat up Tiagra left shifter). No clear photo of front mech (is it Ultegra or something else?). In the photo of the seat tube (not sure why this photo was even included) what is going on with the under side of the top tube? Is this just a weird reflection or some kind of scratch/scuff?

I don't know about the chainrings, but if the chain is worn to the point where the bike isn't shifting correctly, it's likely the cassette also needs to be replaced.

At a minimum I'd be doing the following:
New Ultegra shifter: $150-$200
New Ultegra chain: $40
New Ultegra cassette: $70
New bar tape: $30
New brake pads: $20
New brake and shifter cables/housing: $40

That's $400, plus there's the matter of replacing those ridiculous tires, and finding matching bottle cages. This also assumes the wheels, freehub, headset and bottom bracket bearings are in good working order, and the seller is including the pedals (and they work).

Good luck.