An exception to the above might be made if the person starting in on the saxophone is coming from a clarinet background, and already has single reed embouchure and soprano clef music reading experience in hand.
Learning to play a horn "right" involves four different skills, each of which has to be habituated to the point that they can be executed without thought. They are, in no particular order:
1) The act of blowing a single reed horn. That includes reed control (with the jaw and lips), reed stopping ("tonguing" with your tongue), breath control (support of the air flow through the horn, and regulating the intake of breath to deal with natural breathing points. That's quite a bit just on its own, and if those skills are not in hand before taking up the sax, then a "learner" horn (a Yamaha YAS alto) is the way to go.
2) The act of fingering a horn. There is a lot there too, but the saxophone is relatively kind compared to horns like the clarinet (with different fingerings in different registers, plus all of those first finger and little finger keys). Once you get your hands accustomed to the "keyboard" (and there are some differences between the saxes in this regard), you then have to get things regulated so that the fingers operate in order.
A BIG EXCEPTION:
One horn that makes it a bit more difficult to transition to most woodwinds is the bassoon. It's mostly due to the folded nature of the instrument, this to bring all portions of the instrument within the grasp of the player.
On most horns, there's an even progression upwards of the first three fingers on each hand, with only the left thumb given any additional keys to operate. On those horns the little fingers operate the keys that manage the shift over the break and so forth.
On the bassoon, the left thumb operates five keys to the right side, and another four (plus a thumbhole) on the left. The right thumb only has to deal with four or five. However, the progression "up the horn" involves prodigious action on the part of the left thumb, followed by only slightly less motion on the part of the right thumb, followed then by a relatively normal series as fingers 6/5/4/3/2/1 are lifted in succession. Plus, the left little finger operates keys running contrary to the way the ones operated by the right little finger.
I've had to teach one person (my son) on sax from the bassoon, and it was frustrating in the extreme. He was facile as hell on the faggott, but could just not pick up on the simplicity of the sax compared to his first horn.
3) The act of knowing what to play. Some manage this through rote memorization of the notes, never learning to read any form of music notation.
(Steve Allen was an excellent example of this. He was very talented pianist (and a halfway decent clarinetist, but he could not read a note of music. All of his mass of compositions were worked out through playing them through by rote, and giving a recording of the result to an arranger who would note it all down.)
4) The ability to do it all in unison with the other performers. A sense of rhythm seems pretty simple to those of us who have already developed it. For those who have not, it is a terrifying concept.
My lovely wife, who is a wonderful, cooperative, willing to learn person in every other aspect of her life, cannot carry a beat to save her soul. Put her on cowbell in a R&B tune, and she will be off the beat in less than two bars. Try to walk in unison with her and you just cannot synchronize your leg motion with hers.
Her ambling way of walking and moving is charming in a woman, but a lack of a sense of rhythm is catastrophic in a musician. Most children start developing this in primary school, and by the time they reach the "musical age", they've been singing together, marching in unison and so on for six or seven years.
However, when playing with others in a group (outside of the old "changes" on the rhythm section approach), you need to be able to associate a mark on the paper with a fingering and pressure control combination on the reed and air column. While many have learned to play without this skill, you limit yourself severely iffen you can't read notation.
All told, there's a lot there to be mastered, and if the prospective David Sanborn doesn't have at least 1 and 4 under their fingers when they start, they should limit their activity to the alto. It's an easier horn to manage (both weight and embouchure) than any of the others, and it's far cheaper to afford as well.
If they have 1, 3 and 4 under control, then a case may be made for a transition to another horn. In those cases, a school instrument may be available (particularly in the case of the baritone), and the player can concentrate on the differences between the clarinet and the sax, without having all of the rest placed on them as a burden.
(The first time that I played sax, I was handed a baritone and the chart (the clarinet/bass clarinet/baritone/clave book for Guys And Dolls, with a show date a couple of months in the future), and told to come back carrying my shield, or upon it (so to speak). I won't claim that I was Jerry Mulligan's equal by the time the show dates rolled around, but my background on clarinet and bass clarinet pretty well covered everything needed to function on baritone save a slight embouchure adjustment and development of my left little finger muscles.)
One thing that I would not do would be to start someone out on soprano saxophone. Even with a clarinet background, there's just too much difference with the embouchure to turn out a good player without filtering them through alto first.