According to Omega, the cases on that model would have been stainless steel and yours are 18K gold. However, I was able to find
an eBay listing for an 18K Ref. 2514.
The hands and dial are different, but the seller mentions that the dial was refinished. Unfortunately, this is the only gold Ref. 2514 I could find, so we have a sample size of one.
Have you had your watches opened by a watchmaker and seen the inside of the caseback?
You may have two
Omega Centenary watches, Ref. 2499. They would be from the third and final production run that were made after Omega began to have the movements certified outside their factory in 1951. Here is a contemporary advertisement:
The dials (with Breguet style “12”) and dauphin hands match perfectly. One bonus on this model is that the dial and hands are also 18K gold. It originally came with solid gold springbars, too.
I own a slightly earlier example of this watch, also with the cal. 343 movement.
You can read about it here. Another example with the “Certified Chronometer” script was
sold on this forum several years ago.
Regarding service, while your watchmaker is correct about scarcity of parts, the fact that they are running is a good sign that they may just need a cleaning and possibly new mainsprings. A watchmaker who specializes in vintage watches should be able to take care of that for you.
I see no reason why once serviced, these beautiful legacy watches cannot be worn with pride.
gatorcpa