You can let your kid watch this whenever. But it might be disappointing when you actually watch the show. First of all, I am going to say things about actors and actresses that I feel strongly about. No offense to the actors, mostly I will talk about the show. Characters and such.
Jessie is not a good nanny. She doesn't watch her kids and often leaves them alone. In one episode she leaves the building with the butler and they are still there. At night. One of the kids is little, and two are troublemakers.
Like you read in the Common Sense review, there is some stereotyping involved.
Emma, the oldest Ross child, is the center of the consumerism warning. Although the family is supposed to be rich and famous, she is completely stuck up. She does not stop talking about the helicopter her parents own. (Helicopter? I know, right? THEY HAVE A HELICOPTER!!!!)
I have noticed that in all Disney Channel shows, the family seems to live in a huge house. Jessie and the kids live in a penthouse, with counts as a huge house, even bigger!
The characters act like the only thing that matters are guys. And looks.
They somehow correspond to make some sort of monster show that your kids PROBABLY should NOT be watching.