A curb weight of 3700lb would be considered fairly average these days. There are several "sports cars" that weigh in at 4000+lb, but then they also have 500+hp. I agree that they should really be called powerful grand tourers (GT cars).
IMO a true sports car should be light and nimble. If it's light enough, it can still have blistering acceleration with "only" 300hp.
At ~3500lb, the S197 3V GT wasn't light enough to be proper sports car, 300hp was barely adequate to make it a muscle car, and a solid rear axle was crude technology in an era of more sophisticated independent rear wheel drive systems, but none of that really matters. You can drive it and fantasize about being a movie hero such as Steve McQueen (Bullitt), Nicolas Cage (GISS), or Keanu Reeves (John Wick) behind the wheel.
It was conceived to capture the soul of Gen 1 '65-'70 Mustangs in a more modern chassis equipped with enough creature comforts to make daily driving a pleasure, while the 4.6 3V had the soundtrack to plaster a big grin on your face when you hit the gas. In that respect, Ford nailed it. As always, Ford deliberately underdeveloped certain aspects of the car to keep the sticker price down, and the aftermarket was only too happy to develop a plethora of parts for owners to conceive their own masterpiece from a blank canvas. You could even decide which movie hero you wanted to be and modify the car accordingly.