I decided to take it on holiday with me this year and noticed that the lower chrono button and internal pin was missing.
If the pusher cap had simply come unscrewed, and the "internal pin" (the screw that holds the cap in place) is still there, then I can see that this is possibly poor workmanship on the part of the manufacturer. Even if this was the case, the watch is out of warranty period, so you have no claim for a warranty repair, but they may consider giving you a break as a courtesy.
Note that I apply Loctite to the threads of the pusher screw to ensure that the cap stays on.
But since you have stated that the internal pin is missing, this means it was snapped off. The screw can't back out and go somewhere inside the watch, since it is within a mm or 2 of contacting the movement. Yes it sucks that your watch is flooded, but this is not Omega's fault. As has been suggested, by your own description of the problem it's likely that the pusher was damaged by an impact, breaking the screw and causing the cap to come off.
Note that if it had come unscrewed, this down not happen immediately, so you would have seen the pusher cap on one button being further out from the case then the other - a cap unscrewing is something that can be caught if you pay attention to the watch.
Although it likely would not have helped here in the case where the screw was snapped off, I do recommend that any watch you intend to get wet regularly should be pressure tested regularly.
All blogs online with the same issue of the button falling of were sorted by Omega for free.
Not all - I have recounted this repair here...
Sorry, but you are SOL on this one, and it will be at your expense unless Omega takes pity on you...
Cheers, Al