As for cornering grip, I can attest that raising the tire pressure does increase grip.
I have a specific corner when leaving the highway which I could take at 52 km/h max in the first months of ownership of my car.
Preserving momentum Wayne Gerdes style
When I raised the pressure to 40 PSI (from 28-ish) I could take same corner at 57 km/h.
I assume high pressure prevents sidewall deformation under lateral force so its contact patch stays in shape.
As for braking distance, IDK. It may or may not be affected. There is no lateral deformation then, so I think there is no gain in high pressure there.
But I'd accept a slightly reduced grip in a straight line if I get better handling for evasive moves in return.
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2011 Honda Insight + HID, LEDs, tiny PV panel, extra brake pad return springs, neutral wheel alignment, 44/42 PSI (air), PHEV light (inop), tightened wheel nut.
lifetime FE over 0.2 Gmeter or 0.13 Mmile.
For confirmation go to people just like you.
For education go to people unlike yourself.