George, It sounds to me like the threads for your chuck or insert are either bunged up or machined incorrectly. I have 11 Teknatool chucks, some G3 and most Supernova and Supernova2. I have inserts for my lathes from several sources. (I hate to change jaws.) The runout is better on some jaws/chucks than others but it is minimal and acceptable. I don't use the locking set screws and have NEVER had a chuck or insert come loose, even with large bowls. (Perhaps this is because my lathes accelerate very gently when spinning up and down and never "jolt" the work.)
i would take a close look at the threads. If you bought this locally, perhaps take it to the store and have them test the insert and chuck with a different chuck and insert. Otherwise, you might take it to a local turner who has the same equipment. I am assuming you have the right insert for your lathe - can't imagine the wrong one even partially fitting (you can check the number stamped on the hex flat.)
Note that as you tighten the jaws on the stock it can easily be accidentally jacked to one side as the jaws bite into and crush the fibers, especially when chucking square stock without a tenon. (Incidentally, you can take advantage of misalignment when tightening the jaws to turn multi-axis pieces.)
BTW, this is how I mount long spindle stock in a chuck: If the stock is too large to mount square or quite long, I mount between centers and turn a tenon on one end, making sure to cut a sharp shoulder and making sure the tenon is not so long it bottems out in the jaws. I usually mount between steb centers, both at the headstock and tailstock. If I plan to mount a turning square without using a tenon, I mark the center of the right end and if the wood is hard, make a small hole (with a gimlet). I put the stock in the jaws and barely snug them, then bring up the tailstock (with the live steb cinter) and advance the point into the center hole. Then I tighten the jaws firmly while cranking down on the tailstock. This method keeps the work centered precisely since it can't shift as the jaws dig into and somewhat deform the gripped surface. I use roughing gouge or skew to round the length then remove the tailstock. I often turn 1" to 3" blanks 6" to 14" long.
If the stock is relatively short or I have more thichness than I need, I don't use the tailstock to center but just eyeball it from above, aligning roughly with the lathe ways while tightening the chuck, first on one axis then when rotated 90 degrees. When I have just barely enough wood thickness (i.e. want to make a 1-7/8" piece from 2" stock) I always use the tailstock to aid in centering - it lets me get very close.
In case you don't know, chuck jaws will grip the best on a round tenon a wee bit larger than the jaw's smallest diameter. This not only grips the best but deforms the wood (crushes the fibers) the least.
Also, I make a mark between jaws #1 and #4 in case I have to rechuck the work at some point. And here's a drilling tip for no extra charge: I always start a hole with a machinist's center drill. These are short and stout and get the hole started correctly even with wild end grain. Set of 5 different sizes are very inexpensive on Amazon (for wood, the high-dollar center drills are not needed.)
While on my chuck soapbox, I recommend to get and use a proper wrench to tighten and remove a chuck. I cringe when I see someone put in and whack the chuck key with their hand or piece of wood. I once bought three used chucks from a gentleman who abused them this way and worse: he used a cheater bar to get them REALLY tight. I've never seen chucks in worse shape. The steel in the bodies was actually deformed at the jaw slides to the point that I could barely adjust the jaws. It took me hours to rework them.
JKJ