I'd emphasize the paint mixing and preparation part, as Hawkeye alluded to, more than anything else. I've got an Aztek 7778 kit and a Grex Tritium. The Grex cost three times what the Aztek did, but won't paint any finer a line with certain paints, and unfortunately your paints of choice are the ones that you're going to have the hardest times painting fine lines with no matter what airbrush you use.
A fine line requires that you be very close to your subject (all airbrushes fire paint in a cone... when someone makes one that fires paint like a laser, they'll corner the airbrush market!), which usually mandates lower pressures and very thin paints. You can up the pressure, but you'll sacrifice control and risk overspray. Acrylic paints, for the most part, aren't as easy to thin and get results compared to enamels or varnishes. Tip dry is a constant threat, causing spattering and the loss of your fine line.
Do a search both here and at other sites (Large Scale Planes, Hyperscale, etc.) for Vallejo Model Air and you'll get fairly consistent reports that it's hard to thin for fine line painting, but goes on great for base coats or painting with masks. Several here have stated Future at 1-1 makes a good thinner for VMA. The problem many point to is that Model Master and Vallejo are water based paints. Alcohol based acrylics (Tamiya and Mr Color Aqueous) have a reputation for thinning down with alcohol much better than Vallejo and MM.
I recently tried some Gunze Mr Color through the Grex, and it's an utter dream to spray. No standing by with a wet cue-tip every 30 seconds to clear tip dry, no spiders, nothing. Thin it down to nearly nothing and you'll spray nearly nothing in a perfect coat. After that I sprayed some through the tan tip on the 7778, and got close to the same results. The Grex is still massively superior for mottled cammo, though, due to the much longer trigger throw and (for me) ergonomics and aim.
I'm suspicious of saying 'airbrush technique' has much influence on getting a fine line, as technique tends to bring up a mental image of physical skill in handling. This would dictate how well you can paint that fine line to get the effect you want, but getting that line in the first place out of the airbrush is mostly an exercise in chemistry. Your aztek will paint a very fine line with the tan tip, but if you want to do it easily you're going to have to smell up the place with enamels or varnishes.