Are you saying that non performance tires have no determination on cornering grip? I am confused.
Grip, yes. Understeer vs oversteer, no.
The original FT86 twins had those same slick tires front and rear, so the tires weren't the reason the back end came around. Tail-happy is about several things - rear wheel drive (check), lots of torque/power (no check), weight distribution (no check), more tire grip in the front (no check), or relative roll stiffness between the front and rear ends (big check for the FR-S, smaller check for the BRZ). As a comp, the Miata has very similar power, similar weight distribution and equal-size front/rear tires, but it doesn't have that tail-happy characteristic.
For the example of how much roll stiffness matters, look at the first gen (2000-2003) vs. second gen (2004+) Honda S2000. First gen had a reputation for snap oversteer (earned - I saw a guy lose one so badly at an autocross he hit a light post the organizers thought was unreachable). The second gen could be provoked into oversteer but was largely neutral in its behavior. Both gens had a 48/52 front/rear weight distribution (FT86 is more like 52/48). Both had the same power, but gen 2 had more torque. Both had grippy tires - there was a little shift towards wider tires in the back for gen 2 but not enough to make a real change. But for the second generation, Honda took a lot of roll stiffness out of the rear of the car - a slightly smaller sway bar, but especially softer springs. They also stiffened the front springs a little. You could still get the car to rotate, but it didn't have the same tendency to swap ends unpredictably.
For reference, see:
https://ast-suspension.com/docs/tech-talk/Setup-Tips-Anti-Roll-Bar.pdf
