Vintage Omega Seamaster de ville authentication

Hi all,
I’m new to this forum so please forgive me if I missed something from the “how to post a thread” guide.
My girlfriend gifted me this vintage omega seamaster de ville that she bought from a private seller in the Netherlands where we live, and I would like to assess its authenticity and potentially its approximate value in case it’s original (if she got scammed we know where the guy lives 😀 )
Information I have:
The strap is obviously not the original one, while the case seems like it could be, but correct me if I’m wrong. The case has a few engravings as you can see, two of which being years (1946,1971) which make the dating even more confusing, while the other is probably a previous owner.
The crystal (as well as the case) has a few scratches, but I guess it’s normal for a vintage watch.
The things that somewhat worry me are two:
1) The Omega logo on the dial is not applied but printed, which apparently was less common although not impossible in these watches.
2) The mechanism is very wanky. I have no idea about the servicing history but the watch GAINS about 3 minutes a day.

Thank you very much for the help!

Thank you for the prompt feedback.
I asked my girlfriend and the original listing from the previous owner included a photo of the inside (the seller is a collector so hopefully he knew what he was doing while opening it)