I don't think that convertibles were chosen because of stiffness of the frame/body. Anyone who's ever driven both 68-75 coupes and convertibles, knows that the coupe is a much more rigid design. Even with the extra bracing in the cowl, the convertibles have LOTS of cowl shake, which the coupes don't.
The ease of installing a roll cage probably had little, or nothing, to do with the choice of body style either. If you look at pictures of Corvettes racing in the 60's and 70's, none of them had roll cages, just a single roll bar and forward brace. Few road racing series required cages before the late 70's. Up till then, the only road racers using cages were unibody cars, such as Trans-Am Camaros, Mustangs, etc.; and this was done to strengthen the unibody, not for safety. Sure, it is easier to put a roll bar in a convert., but just how much harder is it to put a single bar in a t-top?
I think the reason so many racers chose the convertible body was weight and aerodynamics.
The convertible body should definitely be lighter than a coupe. Without a softtop frame or the extra steel found in the coupe's birdcage, the convertible would likely be lighter. Even with a hardtop installed, the convert was probably lighter than a coupe, and would be stronger than a convert, without one.
From an aero stand point, I think the convert would be better with or without, a hardtop. In endurance racing, most Corvettes ran a full windshield, and a hardtop. I think the fastback rear window of the hardtop, would be better aerodynamically than the coupes tunnel window. The hardtop's rear window would allow a smoother flow of air over the top, and to the rear spoiler.
Until the late 70's, SCCA club racing allowed convertibles to race with a cut down windscreen, instead of a full windshield. The smaller frontal area of a convertible, with a cut down windscreen, is far more aerodynamic than a coupe's full windshield. Coupes were required to race with a full windshield, putting them at a disadvantage. Around 1980, the SCCA changed this, requiring all new cars that were being built to run a stock windshield. Older converts, with cut down windshields, were "grandfathered", allowing them to keep their smaller windscreens. A coupe might have been more equal aerodynamically, with the rear window removed, but the sanctioning bodies required it to be inplace.
It's true that convertibles were much more readily available in early 68, but this was only the case for a few months. Certainly it's true that race cars get rebodied. I know of 65's that got rebodied as 78s, and later 84's. We rebodied the Trans-Am series Camaro I crewed on in the 80's, twice in 3 years. Still, this doesn't mean that all Corvette racecars started out as early 68's, that were continuely rebodied. Every year new race cars are built, and every year others are rebodied.
Before we built the Camaro in 84 (which was a tube frame, fiberglass bodied, "silhouet" racer), we built a Corvette for GT-1 racing. This was in 1980, we built it from a 77 Corvette, but chose to body it as an 80. We concidered building a convertible, but as I said earlier, the rules by then required a full windshield on all bodystyles. We felt that a 78-80 coupe with a big, plexiglass rear window; would be lighter and more aerodynamic than a 68-77 coupe body and more rigid and similarly aerodynamic to a convertible with a full winshield.
IMHO, I believe that early on the convertibles were more popular than tunnel window coupes, because they were lighter and more aerodynamic.