I was doing some try-outs this evening. I think that part of the problem is that I've become used to quite a bit of rest between sets
This is a major problem with me. I still lift at home, I find it more convenient, while doing cardio at the gym because I can use the pool or the elliptical or the bike or whatever I want. And, lifting at home, usually the lobotomy box is playing in the background and I find myself going from a 60 to 120 second rest period between sets to 5 minutes or more because something on the tube catches my attention. My solution was to use my iPhone's stop watch function and just set it for 2 minutes. When I rest, I start it, and when it goes off, I go to the next set. It helps me keep things in line.
I think I'm just going to have to reduce my rest periods & as you said, accept the time constraints of the system.
You could also rework your routine. I find that people these days do a whole lot of exercises when a few would suffice. It's like Arnold's volume routine permeated everything like a virus and now people think anything less than 40 exercises per body part is insufficient. About ten years ago I 'got back to the basics' and never left. Every workout I do is a variation on 5x5 these days. The most common routine I do is a two day split, day one is Squat, Bench, and Row. Day two is Deadlift, Clean and Press, and Back or Front Lat Pull Downs or some variation like a wide grip pull up. Right now it's Squat, Bench, Row, with day two being light weight Squats, Deadlifts, Clean And Press, Barbell Curls and Triceps Extensions.
I always recommend people take a look at their routines and trim the damn things once and for all. Few if any of us are going to be even mediocre body builders or power lifters, we don't need overly complex routines with numerous exercises to 'hit every angle' or any other such BS. If health and functional strength is your goal, simplify the routine to the major lifts with as few auxiliary/assistance lifts as possible and focus those on any perceived weak points, progress the weights over time, and taking a break every now and then, are all you really need to do.