We started out with thirty or so cows on a flat barn using the big round Surge buckets. So didn't get too much use out of a stool. When we had to, we squatted down between the cows.
And no. you didn't put your head in the cow's flank to keep your balance or anything .
You put it there because it would give you an early warning of when a kick was coming.
Also kept the tail out of your face.