Vintage Geneve Dilemma’s

Hi guys, I'll give you a bit of a backstory here. My uncle recently went to a second-hand shop and bought this vintage Omega Genève Automatic. He asked me to check whether its real and if it is, he told me I can keep it. I believe the reference number is 166.0168 and it should be from the early 70s. I've seen similar models with the blue dial ranging between EUR 500 - EUR 2.000 on Chrono24. I already had the glass polished and put on an aftermarket leather strap because both the glass and the strap were messed up.I've included some pictures below. I basically have two questions:

1. What are your thoughts on the authenticity of this? Watchmakers / jewelers have given me mixed responses, but most of them say it's authentic, the main reasons being that (i) the engraving on the case back looks legit, and (ii) it was never really viable to duplicate watches like these, since it's not considered a "popular" watch to the broader public. When I take a closer look though, the letters on the dial look a bit off-ish and are not clean. It could also imagine a fifty year old watch make the letters go a bit blurry over time.

2. It wasn't running when I got it. I took it to a guy claiming to be a "premium" watch servicer at first, but he wanted to charge me EUR 600+ for a service and to get it running again. That obviously didn't seem worth it, so I took it to a less pretentious watchmaker instead. I left it with them for a few weeks, they said some mini screws in the movement were bad so they replaced them. They tested it for 24 hours to see if it ran, and it did. After I picked it up today, it doesn't run consistently, and stops running every time. When I give it a bit of a shake, it runs again but then stops when I hold my hand still for a minute.

a. Am I being impatient with this? It feels like I shouldn't shake the thing for hours or days to get it running consistently again. I'm not an expert though, so could be wrong. Or should I get one of those watchwinding boxes?

b. Are there reasons to believe there was more wrong with the watch than just those two screws? They told me it should be fine now as long as I keep moving, but I doubt it.

c. Is this a watch I can also wind by moving the crown in 0? I tried it a few times and it doesn't seem to do a lot.

Sorry for the bible verse guys, just wanted to make sure this was as extensive as possible so you might be able to help me out. FYI - I am aware I should not rely on people that advise me with regard to authenticity online, I just wanted some opinions.

It’s definitely genuine - I have the same one. Mine was passed down to me by my father; he wore this watch for a good part of his life and I am fortunate to get to continue its journey.

I paid a fair sum to have it properly serviced by a professional. In my collection, this piece holds a very special place so the investment into the watch was worth it. I also think it’s a beautiful piece.

Thanks alot guys! Very helpful