In the days of the ice man delivering for people ice boxes at home, sections of ice including from icebergs were cut with saws. Ice was stored in ice houses with sawdust or straw for the warmer months.
My Grandfather was born in 1909. He as a kid helped cut blocks of ice from the lake in our town in Winter, sawing them manually like you say. The rest of the year he helped deliver ice, coal, and kerosene by horse and wagon.
I remember my Dad telling me about the iceman delivering ice in a wagon when he was a kid. To his dying day, he called a refrigerator an "ice box." He would have turned 108 today.
One of my favorite Three Stooges episode was them delivering the ice.
The other one was the Census takers.