the experience i have had, with old tires, vs new tires, is for R1 agritculture tires. and R4 industrial tires. 50% tread on old vs full 100% tread on new tires. it was a night and day difference.
the deeper lugs on the new tires, sunk in that much further and gave a bigger / deeper bite into the ground, that kept me moving. vs spinning the tires.
but that is for tractors...
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skid steers are completely different monsters.
and end up seeing more folks recommending tracks that fit over the skid steer tires. for more traction here on TBN.
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you might try "tire chains" for some extra traction in mud / snow. the chains help keep mud/snow from packing up against the tires and turning them into inner tubes. vs tires that have a tread lug pattern to them.