As Crab says, they dont normally just "up" the weight without, for example, beefing up the landing gear. I have no idea what a post- overweight ops inspection might include, but it may not be purely visual. Maybe they would re-examine HUMS data, do some NDT on particular components etc.
SAS - I agree completely that there is something even more worrying in the SOPs if the aircraft are dispatched overweight, but for the airframe, you cant just treat it as if nothing has happened.