To make progress, you need to stimulate muscles deeply. One set typically cannot recruit all fibers in the muscle. That's why you do more sets, to fatigue some of them, then recruit the rest in subsequent sets.
To gain strength, you need to lift around 70% ... 90% of your 1RM (one-rep-max - the maximum load that allows the complete execution of exactly one repetition). On the lower end of that spectrum, that means 8 rep sets. On the higher end, that means 3 rep sets.
I'd be fine with doing one set of 8 reps at 70% 1RM. But it just doesn't sound right to only do one set of 3 reps, even if it's 90% 1RM. There's no way you're stimulating the whole muscle. Lots of fibers will remain unrecruited, and will not be pushed to improve.
Perhaps you're stimulating the same muscle with several different exercises during the same session, but then you're not really doing "one set".
All this boils down to: you need around 24 ... 25 reps per exercise to really dig into all fibers. Shorter sets = more sets = higher load. Longer sets = fewer sets = lower load.
For whole body training, this strategy leads to 40 ... 60 minutes required per session to stimulate all major muscles. Given 3 sessions per week, that means 2 ... 3 hours per week of weight lifting. That's not excessive by any means.
Basically, I consider all of the above as the gold standard of whole-body strength training for amateurs. Using these principles (and others) I transformed myself from a pencil-neck into the guy who bench presses 100 kg (220 lb).
I agree that a full 24 ... 25 reps (3x8, 8x3, 5x5, etc.) program is not required for maintenance only.
To get into shape... well, I guess it depends on the temperament. I'm inclined to achieve the goal as quickly as possible, then focus on something else. Hence my previous reply.
Training programs for beginners are notoriously effective. I could bring a beginner's squat max up by making them walk up hills.
One of the major problems with the scientific literature on weight training is that it tends to take untrained males of college-attending age and then do dippy stuff to them. And then, surprise! The weird training protocol works! The happy researcher concludes that their pet theory is proved.
But as I say, beginners can be trained with almost anything and improved. So one might as well get them used to what they will need to do as an intermediate or advanced trainee and teach them to squat, deadlift, bench, overhead press and (in my case) the snatch and C&J.
For example: a beginning trainee can use almost any scheme and make progress.
An advanced trainee: not so much. Their program will need to manipulate intensity, repetitions, volume, density, frequency and exercise mix to continue making gains.
For intermediate to advanced lifters, a good reference is Prilepin's Table, which gives set/rep/totals based on intensity (% of your 1-rep maximum for that exercise).
For example, when I do 95%+ singles, I aim for 6 or 7 singles.
For 90-95% doubles, I aim to do 4-6 sets.
As I've gotten older (I'm only 22 now, but I started lifting weights when I was 12 so it semi-counts) I've found that I get way more benefit from a bodybuilder style workout. Specifically, rather than work my way up from 5's to triples to singles, I'll do 2-4 sets of 8-15 reps per exercise. I've found that it allows me to target the muscle and avoid strain on my joints.
The reason I bring that up is because I wonder how that compares neurologically to low volume high intensity workouts such as the ones you're doing.
1. Central nervous system adaptation; developing the ability to recruit the largest muscle fibres, to recruit them more completely and to recruit them faster.
2. Myofibrillar hypertrophy; causing the myofibrils, the structures that generate force, to grow larger. (It's possible to provoke sarcoplasmic hypertrophy, which causes fluid build up in muscles -- it grows larger but not stronger).
I am unfamiliar with the physiology of exactly how the CNS produces its adaptations, but I would be surprised if there wasn't measurable effects on the brain, if only the parts related to motor function.
Bodybuilding-style training -- high reps, isolation movements -- is generally easier on the CNS, but has only limited strength development potential.
(For an enlightening read, get a copy of Zatsiorsky & Kramer's Science and Practice of Strength Training 2nd ed).
Once you reach your goals, feel free to switch to a more diverse regimen for maintenance.
EDIT: 15-rep sets is not "bodybuilder's workout". It's a waste of time. Either lift heavy, or go outside and do cardio. That in-between never-never land should be avoided.
EDIT2: Let me restate this. Above a certain set length, weight lifting ceases to train strength - which is its main external goal. The opinions differ, but in general the limit is estimated around 10 ... 12: if your set is that long, the weight you're using is too easy. Use heavier loads, so that you cannot do such long sets. In practice, the longest sets that are profitable are 8 reps.
If you're worried about strain and injuries, do warm-ups with 67% the working load - a short set of a few reps at 67%, maybe repeated once if things are really sketchy with your joints, should be enough. Then lift with a powerful, but fluid and controlled motion. Never jerk the weight.
A serious bodybuilder doesn't give up heavy work though.
http://www.nsca-lift.org/HotTopic/download/Single%20vs%20Mul...
http://journals.lww.com/acsm-msse/Fulltext/2009/03000/Progre...
Aside from the fact that the tradeoff with doing 1 set is actually much less performance improvement long-term, the main problem with this idea is that it just doesn't take all that much time to bust out 3 sets vs 1 set of something, at least for the more critical, whole body exercises (squat, deadlift, bench, chin, row, clean...)
Factoring in generalized warmup, cool-down, travel to the gym, getting changed in and out of exercise clothes, showering, etc - you're talking about a 10-15 extra minutes of training in order achieve substantially greater gains.
For any given action which is physically possible, you're likely to find someone shouting from the rooftops that it's the greatest or only way to train, and someone else reviling it as either worthless or even detrimental. Quite often both contradictory positions have huge numbers of fans.
However, the strength sports are different. Powerlifting requires training for maximum force production. Weightlifting requires training for rate of force production. Strongman requires training for strength endurance. These are three related but distinct physical qualities, and naturally, specialists in these sports train differently. To the amateur eye it can seem confusing and there's a temptation to declare that one of these is "best" and that other methods are dumb.
Otherwise, most of the fad arguments take place in bodybuilding, where the degree of physiological literacy is ... lower, on average.
... on the not-so-glorious side of the ledger, they probably know more about steroids than anyone short of an ex-East German sports scientist.
A weekend warrior may wish to optimize performance/effort, but a serious athlete must optimize performance.