As for double axels, I see this in my rink all the time. There's a few skaters here working on double axels when they haven't landed double flips and cannot even rotate double loops. I don't really get it at all- if the skater can't rotate a double loop, how are they going to do a double axel?
I don't know if it's a region thing (the other rink in town does the same thing) or if it's a trend across the entire skating community.
I can answer from my kid's perspective: because the axel was a comparatively easy jump, and the double axel came easier than the 2L, and it's a big point gainer. Same way the 3T came easier than the 2L
Every skater is different ... the axel takeoff is for some skaters easier than for others, and the toe jumps easier than the edge jumps. Is it a great idea to do the 2A and skip the 2L? Probably not. But, if you're going into competition season, and you need to rack up the points, you'll get that 2A under control and pick up your 3.3 points, and shelve the 2L until you are back to a training period. I also know a few kids who land solid doubles and still can't land a decent axel ... it's partly mental ... partly physical ...
Back to the original OP. I don't see why, theoretically, you couldn't just do an axel and nothing else. If you focus solely on that jump, and develop the technique, you could do it. As the axel is already a "confuser" jump (forward takeoff, 1-1/4 rotations) - it already is different from the singles and the doubles. By working on a steady diet of waltz jumps, backspins, etc etc etc - sure you could do it. In fact, it might be easier as you would be only working on "feeling" 1-1/4 rotations instead of 1 rotation? YOu would also not be having to learn the different takeoffs, again, reducing the confusion of learning multiple takeoffs, landings, etc. In terms of the 2A: same argument, particularly if you've focussed on getting a really solid 1A. It's often a "confuser" for the doubles, as the skater ends up overrotating the doubles due to the additional 1/2 rotation in the air on the 2A.
I would think that any skater who decides to focus on, train, and specialize in one jump could achieve excellence in that jump ... why not? But, it wouldn't make sense, really ...
Same way you could learn a wicked camel spin without learning a scratch or sit spin, I would think.