A quick check of the specifications (https://astronomy-imaging-camera.com/product/asiair-pro) show that each of the power outlets is capable of supplying 3A at 12V.
There is a max 12V input current. However, it is not stated what the total output current is actually supported. The specs show 12V input being between 2A to 6A. The 2A is presumably the consumption of the ASIAIR itself, that leaves about 4A to be shared by the four outlets, perhaps.
ZWO had mentioned the ability to PWM at least some of the 12V outputs (to control dew heaters). It so, those outlets are not purely passthrough power ports. So, in addition to the current carrying capability of the 5.5mm/2.1mm axial power connectors itself, the PWM circuit can be limiting the output currents too, unless you start to add large heavy heatsinks.
In any case, the gauge of the cable themselves are also limiting - anything like 6A over a 10 foot run already requires a heavy power cable. At just 6A, 10 foot run of a 14 AWG pair already drops the voltage by about 0.3V. And I would bet that many amateurs are unaware of this and use cables that are thinner than 14 AWG. This undoubtedly causes some of the odd failures that you read about.
I myself use a PowerPole distribution box to split off the primary 12V power (PowerPoles are capable of passing dozens of amps), and I have a small box with 5.5/2.1mm connectors (a couple are homebrew, and a couple are from deepskydad.com) to distribute power to small accessories that is mounted for example on a dual saddle plate.
For what its worth, deepskydad's DC Hub can support up to a total of 10A power output (each of the five outputs can support a max of 10A, and are short circuit protected by self healing fuses, but no PWM ability).
If you insist on buying an ASIAIR Pro, but need to support more power than the ability of ASIAIR Pro itself, just go buy the DC Hub from deepskydad, or build your own.
Clear skies,
Chen