I've never heard of anyone doing this among any Italian Americans I know. Is this something you've seen first hand?
Why Italian Americans instead of just normal Italian? Aren’t Italian Americans just regular Italians? Or are you asking about the customs of Americanized Italian families or people who were born and raised in America but with Italian ancestry?
Haven’t heard of forever sauce but my family makes sauce by cooking it pretty much all day. If we ate it every day then yeah we might as well keep a pot on the stove all the time