Tire pressures are indeed critical in vehicle handling, especially “directional stability.”
Learned how much so years ago racing sports cars. Proven bringing some firewood home in a bud’s truck. We got on the 2 lane country road after loading up and he had to use the entire roadway, it was so squirrelly. That was at only 35 mph. Stopped at the gas station at the main road and found 20 psi all around. Took the tires up to max sidewall pressure and truck was fine even on the interstate.
My X5 BMW calls for 33 front and 41 rear and at those pressures tows like a dream. A bud who tried it with 35 all around in his X5 was convinced the X5 was a horrible tow vehicle until he tried it at 33/41, when it ran like a train.
Seriously, first thing to check before springing for anti sway is tire pressure.