I can't speak for the OP, but I do check serial numbers - in fact one of the first things I do when a watch comes into the shop is put the serial number into the Omega Extranet. Serial numbers below about the 50 million mark don't work, but most above that do, so this is primarily for more modern watches.
I can tell you that the results are sometimes interesting, and the watch that I have in my hand, isn't always what the serial number would indicate. There is one seller who I've had many of the watches he sells end up on my bench, which is a real testament to the "watchmaker" he uses and the quality of work done. I've seen the invoices and they are getting their full $150 worth for a full overhaul.
🙄
The work is terrible, but that's not the only problem. This seller's watches very often have been modified to look like a different model than what they actually are, and the only way to confirm that is using the serial number. I've had at least 5 watches from this seller that were not the watch that they appeared to be, and had been modified from the original. I always tell the owner right away, so they can then decide if they want the watch returned or if they want to proceed with the service. If I know who the seller is I ask for the serial number before they even send it to me.
Note that you don't need the Omega Extranet access to do this - you can simply call Omega with the serial number, and ask them to tell you what the watch reference is. Anyone can do this if they have the serial number, so if I was buying a modern Omega, I would want the serial number to check to make sure it is what it says it is, before I spent the money. This would be particularly true if I was buying an LE that is known to be commonly modded, like a Japan Racing Speedmaster, Mitsukoshi Speedmaster, or a Mauri reduced, etc.
Yes, from a reputable seller you can always send the watch back, but it would be a lot more convenient and save a lot of time if the seller gave up the serial number so all this could be checked ahead of time.
Cheers, Al