Is there anyone living in Tahiti or Bora Bora on this website? Because your punching technique totally SUCKS and you don't know how to do anything!!!
I prefer movement based drills for warmups (I don't teach the beginner's class, so that would be rolls, shrimping, etc). One 3 min drill is just for movement and warmimg up and then one 3 min drill based on the day's class theme. I also think schools should be open 15-30 minutes before class for those who really want to warm up. I have seen warmups used as a time waster. Also in the more competition bases schools warmups are used for conditioning...in some of these schools the warmups have the opposite effect. Students are in great shape and tough, but their techniques suffer.
Being in great shape and "tough" (whatever your definition of that is--I'm referring to mental as well as physical resilience) can only have the opposite effect. Conditioning can help the frequency, intensity and duration of training--which includes acquisition and practice of techniques.
When well-conditioned (and I do not mean over-exhausted), one can practise techniques longer, with the necessary intensity, more consistently and more often before hitting that gassed-out and/or burned-out phase where mistakes (and possible injuries) can occur.
On the mental side, being out of shape can increase the chance that the participant, overtired due to lack of past conditioning, will attempt a technique in a mental state of frustration and do it inappropriately (example: trying to use muscle in a situation where leverage is the optimal tool). The possible implications, for that individual and any partner, are readily apparent.
There will obviously be some adjustments needed, to any individual's regimen, based on pre-existing abilities and requirements. However, technique-versus-conditioning is a false dichotomy. Well-rounded MA training (for hobby, any level of competition, or due to job requirements) requires both.
I can't speak for grappling arts, but I will opine here that in karate if you're spending half of class time doing calisthenics, your instructor doesn't know how to run a class.