Rolex Oyster dated 1937 - opinions please

[GarryT]

General and specific advice requested please.
I’m really interested in buying the Rolex Oyster (pic attached).
The inscription on the back dates it as 1937.
The seller is my local AD who is selling it on behalf or a client.
The person who knows most about the watch is on holiday at the moment, but I was told in the shop that the watch is totally original – case, movement, hands, face, and that it has just been fully serviced by them.
I’d welcome opinions of whether the watch looks ‘right’ – I’m not questioning the AD, but I’d like other opinions. I didn’t get them to remove the back for a look inside, but I would if I decided to purchase.
My concerns:
The watch is being sold on behalf of a client – so I assume no proper warranty.
If it needs a new part then I assume it will cost a lot to repair (assuming a part can be found).
Is this a money pit and should I just steer well clear?
What are the chances of me having no problems within say five years?
Is it just not worth the risk?
They are asking £2750.