Yes to all the above. Another benefit: it can allow lower overtones to creep (to varying degrees, which can be controlled) into the higher notes: you can do octave multiphonics pretty easily that way, but more subtle effects are possible. It's the growl without the growling.
I agree it helps intonation in certain spots, esp. a2.
This is a very old thread but I'm reviving it because many of these people don't seem to post much anymore and it would be interesting to see what people on the forum now say about the subject and if JL, Dr. G., Gary and PPT, the OP, still do this.
I've gotten more and more into playing without the octave key myself for a number of reasons. First it has a different sound to it when you overblow to the first overtone, because being a harmonic there are more partials in the note and that adds a richness to the sound that is often one I want. Secondly it's a lot faster and easier to play octave jumps on a note in either direction this way than using the octave key because there is no finger movement. Third it allows for a contrast in the two types of tone qualities when playing a line if you can switch between the 2nd harmonic and a note played with the octave key.
So for example you are playing a series of 3rds and you switch back and forth between playing them with the octave key and without it. Repeating a riff that way gives it an added effect or impetus to it I think. Finally, if you are already overblowing for the second harmonic to get the second octave then it is easier to be able to overblow the next harmonic and thus have a larger palette of notes to work with than you probably do when using the octave key because you are thinking of the scale of the horn as ranges controlled by the buttons and not sounds that you can get at any moment using your embouchure.
I don't know if that makes sense, but let me give an example. When I'm playing the second octave G I can play D above instead just with my embouchure. Being able to do that at will allows me to play D3 without needing to use the D palm key thus freeing me from playing a key that is traditionally the way you get that note. Going further that's the whole basis of altissimo fingerings which allow you to play a higher range without actually playing keys up at that range. I am not great at it yet by any stretch but I see that being able to play the harmonic series on as many notes as possible allows you to play every range of the horn with just the keys in the upper and lower stacks and forget about the front F and palm keys entirely.
I play without the octave key sometimes but once I go above G2, then the sax goes back down the octave.
As you keep working at it, you learn to overcome that and increase your range and ability to stay within that register without the key. At first the same thing happened to me. But now it only happens occasionally when I lose concentration on my embouchure. Otherwise I can play the upper stack notes up and down and inside and out at will without the key. It's a liberating feeling in fact and opens up vistas of new possibilities in your playing and not just working towards playing altissimo either.
......................................
The comparison for me to this is the trumpet. I just started to learn Cornet back in June and while I spend way more time on sax than it, I already see that this is possible with the sax because that is the whole technique of trumpet playing. There are only three keys and with those alone you learn to play the natural range of the horn which is 4 octaves. On the sax on the other hand, the natural range of 2.5 octaves is playable totally with the keys either with or without the octave key. This is why trumpet players get pisssed off when they first play with a saxophonist and learn that we have a secret button that does two thirds of the work for us while they're blowing their brains out trying to just play the same range well.
As of no on coronet (and I admit to lazing off during the summer) I can play a chromatic scale from Bb 1 to G1 although the closer I get to G the crappier I sound. Plus my intonation is shaky so far. So I am just trying to get a solid note with core and body to it and hold it for at least 12 beats to begin with as the first half of the goal for now. Pulling in my tonal accuracy is the second half of that goal.
On sax this was nothing to achieve at the beginning, but to do it on trumpet you are repeating some of the same finger positions but blowing a note a different interval above or below based on the interval series. In that simple partial scale Bb & F are both the 1st finger down only, C & G both no fingers at all, and B & F# are both the 2nd finger only. So to give a picture of the notes you can play on trumpet with the same fingering position, with all 3 fingers up and none depressed you can play:
C1, G1, C2, E2, G2, C3.
Thus the sequence is a 5th, 4th, M3rd, m3rd and a 4th. all achieved with the embouchure.
I am mentioning this because we can do the same thing on the sax and have both options of playing whereas the trumpet players only have this one. Therefore IMO, to not practice this method of playing not only misses, but also avoids an entire way of maximizing what one can do with the saxophone to make music.
Just my opinion of course and you may think otherwise, but I think it is well worth the time and effort to learn to do this better and better as I work on other areas of playing too.