The advantages of the 9mm carbine as I see them are:
Shorter weapon.
Adequate terminal performance.
Less flash and noise, though still admitted too much.
I don't know that a 9x19 carbine is per se shorter than a carbine in 5.56, 300 BLK, 7.62x39, etc. Really you need to compare specific weapons. My Aug is shorter than some PCCs I have. My sub 2000 (which I wouldn't use as a primary defensive weapon BTW) is shorter than my 16" ARs. My SBR AK is very small and MUCH smaller than say a beretta storm with a 16" barrel. Obviously some of these comparisons are apples to oranges, which is my exact point. Caliber isn't nearly as determinative of length as other factors.
If you compare similar configuration guns a 9x19 may or may not be slightly shorter. An MP5 for example is (in similar configuration) basically the same size as an HK 53.
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A 9mm Tavor is the same size as a 5.56 tavor. In an AR platform given similar barrel lengths the 9x19 is going to be the same size as an intermediate rifle caliber. I would be willing to go with a shorter barrel in 9x19 than I would in 5.56 or even 300 BLK (or 7.62x39). In 5.56 a 10.5" is as short as I would go and you do pay a penalty in terms of blast and noise. At CQB ranges and with out constraints on ammo selection any velocity loss is not terribly disconcerting. With a 300 Blk I'd use a 8" gun without much concern. In fact for use inside of 200 yards being shorter and having good terminal ballistics is one reason to pick a 300 BLK over 5.56. I've heard of people going as short as 6" on a 300 blk but have no experience and thus can't comment. I've used a 7.5" x39 and it has a lot of flash with the ammo I've used. [As an aside, for all the discussion of caliber no one has pointed out that ammo makes a huge difference in how much flash you get. Load selection can dramatically alter the amount of flash be it from a handgun, PCC, intermediate caliber rifle, or a 308. This point hit home in a low light class I did with various ammo in the same caliber being used.] With a 9x19 you can use even a 5" barrel. In fact for a gun that will have a suppressor you can wind up with something the length of a similar gun with a 12" barrel.
For all the back and forth and frankly uninformed noise, red hearings and straw men in this thread I think it has multiple times been boiled down to the right question:
"Do the advantages a pistol caliber offer make up for the ballistic disadvantage?"
In answering the question I think it is helpful to know if SBRs and Suppressors are one the table. Whether SBRs and suppressors is on the table changes the discussion a lot IMHO.
I also think it is useful to discuss the actual weapon type one is considering.