Did I expect too much from Omega?

True enough. It's not just Omega either.
The watch was supposedly and by accounts taken care of by Omega under the terms of the warranty and as a consequence the customer is typically and customarily not privy to the scope of work undertaken.
In other words, It's none of the customers business whilst under warranty and any work is undertaken at the manufacturers sole discression.

The only real issue here seems to be the claim that the OB's manager has painted himself into a bit of a corner by committing to furnish the OP with something he can't actually provide at all.

Frankly, it was absolutely none of his business where you got your watch or (especially) what you paid for it. You should have just told him that it was a gift and that you're not in the habit of asking people what they spent on your gifts.

In the US, there's a reasonable limit on how many times a new watch (and just about any other consumer product) requires a repair without being deemed a breach of the implied warranty of fitness for its intended purpose, regardless of the terms of the express warranty. I don't know exactly what that limit would be, but the standard would probably be ordinary reasonableness for a watch. If it needs to be repaired 10 times (for example), the manufacturer can't refuse to replace it just because replacement isn't in their warranty.

(It's the exact same underlying concept as lemon laws for vehicles: lemon laws merely codified and standardized the limits of reasonableness that already existed as a matter of law, but mainly so that each new-car buyer who bought a defective vehicle wouldn't have to argue about what's reasonable and undergo the hassle and expense of suing the dealer over a defective new vehicle.)