This is true, your body and it's systems as a whole can accumulate fatigue. Systemic fatigue won't necessarily impact individual muscle recovery but it can impact progress in the long term because your fitness is being masked by fatigue and your workouts aren't as productive because performance isn't what it could be. Good routines will have measures in place to allow the athlete to dissipate fatigue at some point. Some are more forgiving then others, and some guys can just continue to work and progress in a fatigued state despite feeling like a zombie in the gym. Managing fatigue is very important if you are training with a lot of volume and intensity is moderate or high.
I'm glad to hear you are giving higher volume a try. I have no problem with HIT style training for bodybuilding purposes of course.
The reason a lot of guys get shit results with a high volume approach and get better results with HIT isn't necessarily because of the volume or even recovery in some cases, it's because the 'high volume' routine they ran was probably really shitty. It isn't really the high volume that makes routines like that ineffective, it's the lack of intensity, low frequency, no periodization etc. AAS can make it viable but that's not really new.
There are other techniques you should give a try if you are going to move away from taking sets to failure that are excellent at promoting strength gains without making a shit load of fatigue - making use of stretch reflex, lifting the concentric portion as explosively as possible, etc...