Thoughts on my First Vintage Watch?

The OP's watch cannot be a 2846 since it's lugs and profile are very much different. It actually looks very much like ref 2869, of which there happens to be a case back for sale on eBay at the moment, albeit somewhat rotten.

https://omegaforums.net/threads/post-your-1st-vintage-omega.4040/page-11#post-236358

I do not have any new idea about the case back and the originality of the watch.

But I do not think one can say a service is required only considering a gain or loose average. The reason is not that 10-15 sec is good (or not) for this watch. It is that the sign a service is in order is rather a change in the average than the average itself. If a watch gains or looses 20 sec every day, it is very acurate though not very well regulated.

Considering this watch, the reason why a service is to my opinion required is that a nice vintage with no service history should always be serviced!