Both arguments have merit, so what do you do?
I'm a one-skill-at-a-time person. I've had this validated by my Master Coach mentor, who likes to teach pattern and power, and then get down to steps and technique. (I was surprised that he and I shared this philosophy, as I can be something of a maverick since I didn't grow up skating.) After having an argument with a technique-first person yesterday, I was very amused when I opened up this month's Professional Skater Magazine from the PSA, and found this quote from Olympic coach Tom Zakrajsek: "...when I was younger I wanted it done correctly right away. It's all going to work in the long run. Patience. There are steps and that process can take a long time."
The one-skill-at-a-time approach isolates the parts of the skill. Each piece is mastered before the skater moves on. What it is not is "cheating" or "teaching it wrong." The end result will be proper technique on a given skill. The technique-first approach teaches each skill as a whole. The skater will get to the same place, and will make the same mistakes early on, but will have a picture from the start of what the skill needs to be in the long run.
Here's how this might look with basic alternating back edges:
One-skill. Start with the backward glide on an edge, with arms in an "airplane" position, and free foot tucked. You might teach this on a circle, or on a long axis, but the object is to get comfortable with the glide itself. I let skaters use a double pump to get going, because the most difficult part of backwards edges is the first push from a stationary position. Once they can glide backwards, I add the arm and head movements, and then the leg positions. The last thing I teach is the initial push. One-skill teachers do not compromise the final product. I believe this approach helps recreational, once-a-week, and not naturally talented skaters achieve success faster. It gives a skater something to succeed at on the first try. Instead of telling the skater "no that's wrong!" you get to say, "good, you've got a strong glide, let's add the arms." Further, it adds a reward to the process- you aren't allowed to move on until you've mastered each piece; the full skill becomes the reward for your hard work, instead of a mountain you have to keep climbing.
Technique-first: You have to explain the entire move: (face down the long axis, put your weight on your pushing foot, lift and turn the gliding foot, and pump to the glide. Free foot in front, drop and switch arms, turn your head toward the long axis, and bring the free foot through, setting up for the next push from the one foot glide. I know I'm showing my bias here, but really, how many FS1 skaters are going to be able to absorb and understand all that, let alone do it?). There's going to be a lot of hands-on guidance, placing skaters in the proper position, and guiding arms and heads. In private lessons, this works, but for group class, it forces everyone to stop while you work one-on-one with each skater. However, each skater is watching you (one hopes) working with everyone else, and absorbing the correction and the look of the skill. You'll get a lot of sloppy skating at the start, but theoretically a better understanding of the end result.
So which do you choose?
What I would say is: don't choose. Look at each skater and make a judgment. Can this skater handle the whole skill at once, or does she need it broken down? Are you sure that by choosing one over the other you are not compromising the quality of the final product? Are you using consistent, correct evaluation? In a class, you might encourage one skater to go for the entire correct technique, while allowing another (generally weaker) skater to break it down and perfect one skill at a time.
In the end, it's not about the teaching philosophy. It's about the best interests of each skater, and the approach that will help them be most successful.
What works best for you?