I'm certified to teach K-5 in one of the US states but currently teach in another country. I've looked into this idea quite a bit.
There is evidence to show that extra school time benefits children in families that don't give much academic support at home (especially prevalent in poor, inner-city neighborhoods). For example, standardized test scores in reading often rise after summer vacations in affluent areas, but not so much in poor areas. The assumption is that many affluent parents tend to read and encourage their children to read during the summer. It's simply a disparity of time reading. To combat this, some experimental, inner-city schools have had success raising scores with very long days. However, I haven't seen anything showing that longer days help elsewhere. Homework (no matter how many hours) has been shown to have no significant effect raising scores for elementary students. (Up to 2 hours helps high school students, but over that seems to give no additional benefit.)
Honestly, I would first look at reducing time giving children tests. In many schools, children are given about an hour of tests a day, on average (amount varying from day to day, class to class, school to school). Tests are specifically to help adults (administrators, teachers, parents). Children are not allowed to practice their weak areas (the main thing that helps them learn) during a test. Although tests give children goals to strive for, motivational goals can be given many other, more effective ways. That's often 180 hours of test time a year (36 days of school, considering 5 hours a day of "in-class" time).
In my school we give 1 standardized test a year, and no testing outside that. Our scores are usually average or better than average on the standardized test (despite having many special-needs students). The teachers have more time to work with the students (and therefore know exactly where each child is). We also have more time to plan (instead of correcting tests during prep time).
Common questions we get are about how we communicate a child's level, without grades (given from tests). Simply put, we give more in-depth reports to parents & other schools. It works, but this is the part that scares most administrators and parents. Frankly, this part is more work for the adults. But if the main focus is on what's best for the children, frequent testing should be abolished. From the perspective of a child's education (practicing difficulties and learning new things), testing is one of the least efficient uses of time. And if we truly want more class time, that's where educators should start.