Aggressor squadrons are essential in training, but I don't see much sense behind them flying aircrafts that pilots won't see in combat. They can use "enemy" tactics, but those are defined by aircraft capabilities, and F-16 is different than MiG-29 or Su-27.
What US pilots would see in the air are Su-27, MiG-29, F-14 and Mirage 3/5/F-1. and none of these jets is used in aggressors squadrons. The F-15 won't mimic Su-27 or F-14, nor the F-16 be capable of doing what a Mirage or MiG-29 can do.
Why the DoD didn't fought about inviting foreign forces to play the aggressor role ? What would be the problem of inviting even a whole squadron of MiGs with pilots who have years of experience, and could perform their role far better than anyone else ?
As someone else commented, tactics are as important if not more so than aircraft capability. Yes, tactics are based on capability, but that just means the tactics employed may not be optimal for the Viper, but they are for what they are imitating.
As for bringing in a foreign air force with the real thing...can't do that. We don't want to reveal our tactics and ultimate "bells and whistles" to others. Our highly advertised "loss" to the Indians a while ago is an example. Yes, we were surprised by a few things they did and their capabilities, but we went to the fight with one arm tied behind our back so as to not reveal stuff...