It is my understanding that SOME convertibles may have come with the cowl bracings. Most, however, didn't. If you see a car with them, they may have been added later on. Both the frames and cowls on convertibles were reinforced, so the braces were not considered necessary. Coupes and sedans had frames made of a lower gauge than the convertibles due to he solid tops welded to the quarters, so the bracing may have been an overkill on these cars. After the factory test drive, a shaky car might have had them put on.
As time went on and the cars had some wear, owners may have added them for additional rigidity.
An old factory man could really shed some light on this! Prior to the mid-60's, there were NO additional braces on any of the cars. Were the "X" frames really that solid? I know my '57 Coupe de Ville is as solid as a rock.