With 275/55/20 tires , at 62 mph (100 kph) your motor is spinning 3,425 RPM in 3rd gear or 5,275 RPM in 2nd gear. At 3,425 RPM you have about 315 hp available, at 5,275 RPM you have about 415 hp. Big difference there. If the trans won't let you downshift to 2nd (not surprising, 5,275 RPM is moving) then I understand why grade-ability could be better at that speed. You could always set the trans to manual mode ("M") and hold it in 2nd at 55-60 mph. Or try to hold 3rd gear at about 4,000 RPM (73 mph!).
If you had the 8 speed/3.23 with same tires 62 mph would be 6,250 RPM in 2nd, 4,400 RPM in 3rd, and 3,575 in 4th. If 4,400 RPM nets about 370 hp on tap which might be enough for most grades, and if you dropped to 50 mph you could probably downshift to 2nd again (5,000 RPM) so I would say the 8 speed is a slight improvement.
For reference, your old 4L60 with 4.10 rear just held 2nd gear at 4,350 RPM at 62 mph. It never shifted into 3rd because that would be 2,675 RPM which would be gutless. That's why it seemed to grade better than your current truck. Just happens to be the ideal speed for the 4L60 in 3rd gear. Ideal speed for your 6L80 transmission is more like 50 mph at 4,300 RPM. Just a bit of an unfortunate overlap of speed.