Tell that to all the S2000 and Miata owners...
As others have pointed out, both of those cars were designed from the beginning as convertibles, which helps chassis rigidity a lot.
However, both models would still be both lighter and stiffer if they were conventional hardtops. The reason that these cars see heavy track use is simply because there really aren't other options available within their preice ranges. The Miata has no real competition, there just isn't another RWD sportscar out there in its price range, and even the S2000, when it was in production, was the only option out there if you wanted a "step up" from a Miata in terms of power.
Because both the Miata and the S2000 were, effectively, specialty cars (a very high number of them are second cars), making them convertible-only was a good move. Marketing wise, convertibles tend to be more popular as "second cars" since many people tend to think, "well, as long as I'm indulging in a car, I might as well have fun and get a convertible." This is why many classic convertibles, like the Porsche 356 "Speedster", are currently far more valuable than the coupe versions of the same car, even though the coupes originally sold for more money when new. Hell, when it was new the "Speedster" was the stripped-down, low-price variant of the 356. Anyway, since it would have cost more to build both coupe and convertible models of the Miata and the S2000, it made more sense to produce only a convertible model (and mitigate, but not eliminate, the inherent drawbacks) and thereby retain greater popularity in the "weekend car" market.
ZV