I think Harriet is useful to Emma not just as a friend near her own age -- and I think we can agree Emma needs one -- but as a "project." Emma is bored. She isn't looking forward to spending all her time with her father's old friends, and there's no Miss Taylor to make her spend her time more constructively. I have to think that even though Miss Taylor wasn't as strict as a governess might be, she did nonetheless have enough influence on Emma to keep her busy and out of mischief. In addition, Emma's respect for Miss Taylor might have kept Emma from getting her own way all the time.
In Harriet, Emma has someone to whom she can be the superior. You'll note that Emma always thinks of her former governess as "Miss Taylor" or "Mrs. Weston," while the latter always calls her by her first name. Here the situation is reversed -- Emma is always Miss Woodhouse to Harriet.