Lots of cars are "sporty" cars, but the Miata is truly a sports car. The handling is superior to many $50,000+ cars out there.
Regarding the height "issue," a 6'4" basketball player at my college owned a Miata and he said he had no problems getting in and out. And there is a ton of legroom inside.
The Miata isn't just sporty, it's a true sports car. Drive a Miata. Take it on a mountain road. Test the handling. Then tell me it isn't a sports car. Your comments regarding performance lead me to believe that you've never driven one.