If you look at the list of positions which can be used to meet the position of responsibility requirement for Star and Life, you will see that assistant patrol leader is not listed there.
Star Leadership Requirement
National Youth Leadership Training (NYLT) is an intense week long leadership training class for youth
Star is the Scouts BSA rank earned after First Class. With Star, the focus shifts from scout skills to leadership, service, and merit badges.
A Scouts BSA Troop is actually run by its youth leaders. See the list of Scouts BSA youth leadership positions.
The Den Chief Service Award recognizes Scouts and Venturers who serve a Cub Scout den for one year, are trained, and are active members of their Troop or Crew. The award is a red, white, and blue cord, which is worn on the left shoulder under the epaulet. A certificate is also available.
While working on the requirements for the Star rank, Scouts must serve in a leadership position for at least four months. Or he or she can do a leadership project approved by the Scoutmaster. This requirement helps their troop and enables them to learn leadership skills.