He changed his clothes because they were dusty and dirty from all of his traveling and the hay he carried to burn the mass grave. But he didn't have to have a bath because his body was clean.
It was true that he didn't sweat through his journey because his body wasn't the one doing all of the work. But even the hands that he used to carry hay were clean. So the cleanliness of his body couldn't be attributed to chance.
As he boarded the airship in new clothes, he smiled and said to himself, "This Pristine Sword Metac is already paying dividends. I hope that things will continue to go this well from now on."
As luck would have it, everything went well from then on. There were no accidents on their way, and they arrived at the world barrier just in time to see the large ball of fire from the sky into the world barrier.
Loki didn't witness this sight, of course. He was in his cabin, busy cultivating.
