I don't think the "manager" and "memberOf" properties are retrieved by default. You'd have to inform the Get-ADUser that you want them by naming them in the -Properties parameter.
However, just because you'll be populating the "memberOf" property of the new user won't, I think, actually make the user a member of the groups. In other words, the "members" property of the groups won't be updated to include the user even though the user's memberOf property would lead you to believe that they're a member. You should verify that.
The same may be true of the user's "manager" property. I'm not sure that simply copying it from the template user to the new user will also update the manager's AD user property "directreports".