Square gates are actually the standard in Japan. I think a lot of people just move the stick around with their fingers rather than with their wrist/hand to avoid getting caught in the gate's corners when playing fighters. The looser feel is probably because that controller has a Sanwa stick in it; Seimitsu sticks are tighter and generally preferred for shooters. I'd be interested to know what you think of the controller for IaMP/SWR, since you're now pretty much set up with the authentic input method for them.
You can buy an octo-gate for the JLF that comes with the TE sticks, but as you mentioned Seimitsu sticks have a smaller dead-zone and throw. The LS-56 has an octo gate available for it but you need the MS mounting plate (the VF that comes with it isn't suitable). The standard Seimitsu stick for SHMUPs (LS-32(-01) comes with a square gate, and you can swap that out for a circular gate if you'd like.
That said, don't ride the gate. The JLF has a long throw (the total distance you can move the stick) and dead-zone, but the engage distance (engage is when the switches activate) is much shorter than the throw. It's a matter of finesse, but you don't need to push the stick all the way to have the movement register. Wish a square gate the distance to a corner is twice as far as to the sides, so it isn't hard to avoid mashing down into a corner.
I find that the square gate makes the rectangular playing field more believable. Should you want to make a big sweeping movement to dodge aimed bullets like the ones Cirno opens up with and just so happen to bump into the world's boundaries, it kinda reflects upon the general shape of the world. This shouldn't come into play often though since they activate far before the ends the edges of the map are generally where you don't want to be. I must admit my first instinct to try and throw all the way out failed rather miserably. Thankfully it's really quite easy to take note of where the switches activate because there's a very satisfying little ticking type sound when it happens. That makes picking up on what you should do instead quite a quick process.
The stick differences are interesting though. I did find myself initially expecting a smaller throw at first and I suppose that in conjunction with a tighter stick, would allow you to go back to neutral quicker. I'm not really experiencing any problems with that though and like I said before, this stick feels very quite natural. Far banking means you go far where as the simplest little finite touches move you in microns. I'd attribute this partially to the amount of time it takes to return to neutral upon, which becomes greater the farther out you go.
Also the finesse issue, it can be a bit of a double edged sword as more is required but more is allowed. While I must admit to having little frame of reference, I'd imagine that the looseness means less resistance to your will, making mid-motion adjustments a lot easier. I must also say that it makes putting in very light touches with your fingers very easy. These factors can be great for getting out of tight squeezes a lot easier. However the results of carelessness are also more accurately reflected as you can make silly decisions less hesitantly and may not be as able to just stop on a dime.
With a stiffer stick, one's touches probably wouldn't have to be as light to register small movements either, which is good should the user prove a bit too heavy handed. It may may help to prevent slip-ups as well since commands are less likely to be misregistered and the greater hesitancy would allow for greater caution. Quicker returns to neutral would also guarantees the ability to make quick stops on the dime. Essentially you're a lot less accident prone, which is very important for a long haul game like an arcade style shoot 'em up, since if you do it all in one sitting you're bound to make some eventually.
All in all I'm probably going to leave it stock as a pseudo collector's item should be, especially since it's quite impressive as is.
I'd be interested to know what you think of the controller for IaMP/SWR, since you're now pretty much set up with the authentic input method for them.
There's a bit of irony in the situation in that even though I got a Street Fighter Fightstick/Pad, I get really disoriented by fighting games due to their control intensive play-style. True, this style of play probably makes them the best barometer of what sort of controller is good or bad but I can't comment on my effectualness without knowing what it is I wanted to be doing. Refined articulation and crude button mashing just simply don't mix. Also keep in mind that this is my first and possibly only high-end home joystick, so I don't have much in the way of fair comparison, unless you count my keyboard/playstation controllers. That being said, I did order IaMP, Mystical Chain and Melty Blood recently, so when I get everything I need from the mail, I might try to toy around with various control schemes a bit. No promises though.
You're more verbose then me, Tonepoet. 
I sent my stick to warranty too, for the stick mode selection that didn't work. But after one day of play with it, I really don't regret it, I honestly thought that it's the best hardware investment I did for the last months (~year).
Eheh, yeah, I do tend to go all out with my forum posts. I think it's a great use of the medium, when done properly, although I do sometimes question my ability to do that sometimes... At any rate, it's not comforting that there are two warranty calls here, although mine did seem to be fully functional. I probably wouldn't be able to really tell if my switches worked or not though, since I've only hooked it up to my computer. Hmm...
