I do not know what AMRAP stands for but allow me to explain how we use Muscular Endurance training and why we do it this way as opposed the use of High Intensity Circuit style workouts like (from what I can tell you are proposing). There is a significant difference that is not understood in the general fitness world which has lead to a proliferation of many different HIIT circuit programs.
I’m partly responsible for some of this confusion so bear with me. What we call Muscular Endurance (ME) is more correctly called LOCAL Muscular Endurance (LME). This was the term originally applied to it by Yuri Verkhoshansky (the inventor of plyometrics) , who from my reading should get credit with developing this protocol. For expediency, I shortened LME to ME many years ago when I was working with individual athletes. I gave no thought to the loss of accuracy. As the popularity of Training for the New Alpinism took off I now realize the confusion my casual renaming has caused.
The type of workout you describe (if I understand you correctly) has a GLOBAL endurance training effect. Our ME workouts are designed to have a LOCAL endurance training effect. The difference may sound subtle but these have a profoundly different effects that needs to be understood in order to use them wisely.
An LME workout is designed to cause the endurance limit to be reached at the LOCAL muscle cell level. We do this by adding weight/resistance/hyper gravity so that what limits your work capacity is a relatively SMALL (LOCAL) muscle mass. But these targeted muscle fibers are THE muscular limitation to your endurance. In these workouts the local fatigue limitation should be so great that heart rate and respiration do not become dramatically increaesd
A High Intensity Circuit workout is designed to cause a GLOBAL endurance effect. The limit to your work capacity there is going to be the cardiac muscle’s ability (hence the name CARDIO often used with this style workout) to supply oxygen to a much LARGER muscle mass. In these workouts heart rate and breathing will reach near maximum levels.
Why do these distinctions matter?
1) HIIT or High Intensity Circuits do have a LME effect for sure but it is limited by the athlete’s global endurance. If any component of the whole endurance package is low then the LME effect will be limited. We have seen for decades that there is a direct correlation between global endurance and LME. Raise an athlete’s LME and he/she will exhibit more endurance. LME is one of the most trainable endurance qualities.
2) HIIT (as in conventional endurance sports) or High Intensity Circuits (Crossfit, Orange Theory, P90x and the like) create a massive disruption to the body’s homeostasis. They make you globally very fatigued. They have a pronounced anaerobic training effect. These are useful benefits when used in a focused, moderated and periodized way and all endurance athletes will see gains with proper application of them.
We’ve devoted many thousands of words in our book(s) and in many articles on this site to driving home the point that the aerobic metabolism is the main driver for endurance and must be maximized to see the best long term gains. Where people go wrong with the use these global high intensity workouts is in there miss and over use due to not understanding that subtle distinction I just explained. These global high intensity workouts REDUCE aerobic capacity and bias your metabolism toward the anaerobic production of energy. Just the opposite effect we want to achieve long term. This throws the aerobic/anaerobic metabolic balance off. We often see people who have a history of overusing GLOBAL high intensity work who have completely detrained their aerobic system. For them it will be a long, slow and often frustrating period to fix the metabolic train wreck they have caused.
Because LME workouts have their main effect locally they do not cause the same level of anaerobic metabolic shift as do the global high intensity workouts. With LME we can continue to build high aerobic capacity at the same time that we are having a profound training effect on the muscle fibers that mainly limit endurance.
In conclusion, the type of workout you are suggestion can play a role in training mountain athletes. Depending on the sport it will factor in more or less heavily. When we use then we use them in a very sport specific fashion rather then in a general circuit. The sled push is a useful tool but I’d structure the use of it differently that you are suggesting.