You don't like the older Mustangs so they aren't sports cars in your opinion. O.K. good for you.
I love Mustangs. I have one. Iv'e had a number of them over the last 30 years. I've even autocrossed a couple of them quite successfully.
None of them were sports cars. GTs, yes, Sports cars, no. Too many seats, for one. I'm a purist, from teh old school. Sports cars are open, 2 seat cars designed with agility and handling for road racing as a primary function. Any more seats or add a roof and it's a GT. Any more doors and it's a sedan (and yes, i believe there are good sport sedans and sport coupes, but they are not what we woudl call sports cars).
Sports car makers know this instinctively, which is why Porsche 911s when they are set for motorsports are not sports cars but GTs, and say so even in the name (GT-One, GT2, GT3) Porsche has been around for the beginnings of this, and they know. So does Ferrari, who take it one step further and decalre any sports car that is primarily a road car a GT, which is how you get a GTS (Grand Touring Spider) at Ferrari.
Nothing wrong with calling a car a GT instead of a sports car, as GTs were usually faster than pure sports cars due to better aerodynamics of the closed roof bodystyle. But look at these cars and tell me the difference (and notice the naming of them), made by one of the oldest pure sports car manufacturers:
There you have an MGB and an MGB GT. The difference being the fixed roof. The definitions still work, it's just that the cars in each category have gotten better (Like the Nissan GT-R. Notice the category). The Mustang is a traditional GT, not a sports car, and the name is accurate (my Mustang says so on the fenders)
Now, just like the term "roadster," "GT" has been grabbed by the marketing folks and bastardized in recent years to be put on cars that have nothing to do with the definition (and a 5 door hatch is not really a GT). But the rule of thumb is, if the company is a sports car company, then they probably know what the definition is and stick to it. BMW is the exception, as they pretty much invented the term 'sport sedan" even though Alfa and Lancia made them before BMW did. So why they call the 5er GT a GT is beyond me.