I think TUT is most important to track but they're all important and help you form a big picture of your progress. I look at my reps and TUT from the previous week and try to improve both. Sometimes I fail with fewer reps but more TUT. This usually indicates that I was moving too fast the week before. Sometimes both improve which is as close to an objective indicator of growth that I can find without spending tons of money measuring muscle volume. Other times both are weaker which forces me to reevaluate what is going on. Did I get enough sleep? Did my body fully recover? Am I distracted? Am I being lazy? It's useful data that can be easily tracked with a stopwatch. You can buy one and keep it around your neck. Start it and then begin your exercise. Stop it after you finish. Write down the weight, time, and reps. Monitor your progress over time. If you just base your progress on how you feel and look, it can be misleading. Some people look more muscular after they lean out but actually have less strength when they are bulky. It could be that you were getting stronger but you didn't get enough rest to fully experience that strength. I sometimes give myself 2 weeks rest just to be sure. You can experiment this yourself. Take two weeks off from training and just see if you feel stronger doing BJJ. If you do feel stronger, and you weren't training, what would be the rational explanation? We don't really have a signaling mechanism in our bodies that tell us when we have fully recovered.

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Morning resting heart rate and heart rate variability right after waking up for cardio system.

Grip strength for central nervous system.

Bar velocity at a fixed weight for muscles.

Those 3 systems are capable of separate independent stress and recovery cycles and need to be individually monitored.

You say more breaks but I'm training more now because I'm doing a kettlebell day in addition to my barbell day. It doesn't take long but it is hard as hell.

Do you know what can help your nervous system recover?

All I've found is sleep and other "remove all stress of any kind for a while" approaches. I like to lay on an accupressure mat listening to relaxing or healing binaural beats on my noise cancelling headphones and do mindful progressive muscle relaxation.