Why are high-g requirements so important for training next generation of fighter pilots?
Two solid reasons: defensive and offensive ACM/BFM remain relevant

I'll address the defensive issue first.  Even if a pilot will never get into a dogfight, the pilot / aircraft still must defeat or evade a surface-to-air or air-to- air missile.  This combines the use of both countermeasures and 3-dimensional maneuvering to cause the guidance system inside the missile to lose it's lock on the aircraft.  To get the turn rate and axis-change-rate needed to deal with modern missiles requires the ability to pull a significant number of g's.  You can look at is as a "purely defensive dogfight" technique.  You fight the way you train, so you need to train for this likely situation.  

The offensive high energy, high-g capability has to be retained because in real life, the rules of engagement do not always allow for a BVR (Beyond Visual Range) engagement.  Further that point, even with BVR engagements allowed, if the opponent has good defensive technique versus your missiles you may still get into a dogfight-style of engagement in order to defeat or destroy that opponent.  You have to know how to do that, and train for how to do that.  It is an acquired skill. (The basics of the skill were eloquently laid out by John Boyd, which is where the OODA loop model comes from).  

Both the offensive and defensive applications of these maneuvers require skill with and acclimatization with high-g maneuvering.  

A related reason to boost the performance of the Trainer

For the last 20 years, it has become fashionable, and sometimes useful, to do a thing called "downloading training" into a cheaper to fly, cheaper to maintain aircraft.  That way you don't wear out the airframes of your very expensive front line aircraft.  (The Navy spent years using the T-2 for spin training for its jet pilots, to include T-2's not used in the Training Command but retained on both coasts as "spin birds").   This concept of "downloading training" sounds nice, but in reality has mixed results.  At the high performance level that front line aircraft operate, some skills are transferable and some are "building block" from one airframe to a more complex airframe. Given how recent this requirement is, I'll see in this requirements crafting something similar to the initiatives that I ran into in the Navy (and in Joint Training) with "downloading training."  The same mind set is at work here: if the spec can be met at a high enough level, the graduate of the advanced jet training will enter the front line units with a higher level of capability.   We successfully down loaded quite a bit of instrument training into the sim in at least one aircraft program I worked on, the T-6.  While that was 15+ years ago, the never ending push to "do it cheaper" has not let up in the DoD. Our efforts to download ACM/BFM into the T-6 for the NFO training pipeline ran into the problem of "not enough energy."  I am not sure if that has since been solved, or abandoned, but I think it's the latter.  (The last time I checked on this was about 6 years ago).  



Note: This is an experience based answer, informed by my having both been taught basic one-versus-one maneuvering, and many years later having been deeply involved in training systems development, and training requirements, for US Navy fighter/jet pilots. (They had the same challenges to overcome as their Air Force colleagues in the proposed T-X system).  A more detailed answer would be, roughly, book length.  Chapters 5 and 6 of the link cover Offensive and Defensive BFM basics.   

From the introduction to that manual:  


  BFM INTRODUCTION
  When the P-80 finally became operational in the mid-1940s, WWII was coming to a close, as was seemingly the age of aerial dog fighting. While these aircraft never took part during operations in Europe, they became an integral part of combat operations in Korea. In November, 1950, a P-80 flown by LT Russell Brown shot down a Russian MiG-15 in the world’s first decisive all-jet aerial battle, acting as a reminder that BFM was still a very real possibility in combat operations. With the maturation of aerial warfare evolving into Beyond Visual Range (BVR), the decades that followed Korea exploded with the development and implementation of long range air-to-air missiles that could potentially eliminate the possibility of Within Visual Range (WVR) engagements. During the Vietnam conflict, however, aircrew often found themselves unable to employ their BVR missiles, forcing weapons employment into the WVR arena, ultimately relying on BFM skills to defeat their opponents. The days of Korea and Vietnam are long past, and we have achieved amazing successes with the accuracy and reliability of our current long-range
  missiles.  As we, and our enemies, continue to improve our BVR capabilities, a combination of electronic attack, theater Rules of Engagement (ROE), switchology failures, or a momentary lack of attention to prescribed air-to-air timelines may ultimately bring us face to face with the enemy in a 1v1 engagement.
  (CNATRA P-1289 (Rev. 01-16))   


The airframe may change, but the problem remains the same.