But what difference do grades make in elementary school? There are no real consequences to lower grades until a few years later. I'd let my DD suffer a lower grade for not completing HW.
The consequences will vary depending on the school and family cultures and the individual's personality.
In my kids' school, everyone knows who isn't getting good grades, and they are in a pretty small minority. Even merit roll is rather mediocre. A child in that environment can easily develop a mindset that she is a mediocre (or poor) student with a mediocre (or poor) mind. That can have far-reaching implications. It also isn't a good idea for the teachers to think your family doesn't give a damn about education. The teachers' view of the kid and her family will determine the amount and quality of attention the kid gets in school.