Seamaster with Scratched Out Movement Serial

Are you referring to all bridges, or those with serial numbers specifically? Not sure why they would want a bridge back that wasn't serialized, because I really doubt that they refurbish them - would be more trouble than it's worth.

Rolex policies with regards to spare parts (like not giving them back to the watch owner when they service the watches) are designed to further the idea that their watches are somehow particularly robust (to the point of being magically indestructible), and that plan has worked well.

When returning any serialized bridges to Omega, if you remove parts from it, they will not accept it - I know this first hand...

Quartz movements of this era were not serialized, so the only serial number on those watches is on the case. If that is removed, they will not service it as you found out.

Something to consider for those who may want to purchase a fairly modern quartz Omega.