Sold my gen 3 Superduke R end of last year. Ownership overlapped with R1300gs.
I actually found the GS to be more stable, feel is different, no surprise really. The KTM would let loose very suddenly, the traction control would catch it...
The GS needs to be in dynamic pro and set up. Do try altering damping too. I backed mine off, which helped with corner grip for me in the Alps. Adjust traction control to minimum, throttle response to max, and yes, fit good tyres . Conti Road Attack the best I have used, Michelin Road 6 were good, but I had to drop the tyre pressures..the rear was starting to slide early when pushed.
Tyre pressure does make a big difference, just like any other bike. Majority of GS riders would just stick to manufacturer spec I suspect.
Warm weather/dry roads I run mine 34 * front 38-40 rear (solo) and cold weather/damp roads 36/40.
*below 33 the steering becomes heavier and kind of drops in at slow speed.
Sticky options;
There is a Dunlop, though can't remember which one
Metzler M9rr
I have not tried them as my bike gets ridden long distances on tours, not sure how the softer tyres will hold up
Conti Road Attack 3s DO start to rip up on hot roads when pushed. (35 degrees and Pyrenees..) but grip and predictable handling is superb.