Next-generation plane engines take a LEAP toward delivery

But neither of the two major plane makers are directly responsible for much of the promised improvement in fuel usage.

This is not entirely true since improvement in design can reduce amount of drag and reduction of weight which will contribute in fuel efficiency.

Would a shock cone in each engine reduce the drag?

Has anyone looked at the supposed more efficient type of turbine that Nikola Tesla patented but was never adopted?

Would a shock cone in each engine reduce the drag?

A shock cone is used to slow supersonic airflow down below supersonic speeds before it enters the engine. As airliners don't exceed the speed of sound, there's no danger of supersonic airflow entering the engine.

Has anyone looked at the supposed more efficient type of turbine that Nikola Tesla patented but was never adopted?

The Tesla Turbine does not do work by itself, and instead must transfer the power generated to some other mechanism in order to do work - similar to how an automobile engine doesn't actually propel a car along the ground, it needs to transfer its power to the drive train so that the drive train can do the work. An axial jet engine in an aircraft does the work of getting the aircraft airborne without needing to transfer power to some other mechanism.