Controlling jet engine thrust on airliners
Are there any attempts to allow faster thrust response than this? Are any in current or planned use?


First, for turbine engines, the more often used term is thrust lever or power lever rather than throttle. You can Google "throttle vs thrust lever" for that discussion.

I'll use the 727 and the 747 to illustrate what was done prior to when I retired in 1999.

In initial 727-100 sim training, the instructor had us bring the power levers all the way back and allow the descent to stabilize in landing configuration at full flaps. Then, at a given altitude the instructor would tell us to start climbing, at which time we would advance the thrust levers to max power and clean up. From that moment, we would lose hundreds of feet before the aircraft started to climb. The primary reason for the delay was that the engines took 7 seconds to go from flight idle to full power.

In initial 747-100 sim training, we did the same thing. The result was a loss of less than 100 feet, the difference being that engine response was almost immediate, and we also could pitch up immediately without noticeable loss of airspeed.

As I remember being told, the big difference in engine response was because the idle position still had a relatively spooled up engine.

The engines running at that relatively high power output even at idle had the drawback that while taxiing the aircraft would accelerate to a higher than permitted taxi speed, which then required braking.