いってらっしゃい itte rasshai

Unfold the contraction and you get itte irasshai, go and come back, with the honorific verb doing double duty for going and returning. The reply to itte kimasu is therefore not goodbye at all. It is an instruction to return, issued every morning at every front door.

Once you notice that, the phrase turns quietly enormous. It builds the round trip into the farewell, so parents say it to children heading to school, shop staff call it after departing customers, and it can wobble a person leaving for something genuinely uncertain. Translation flattens it into have a good day, which is friendly, sensible and missing the whole point.