What determins the available amps in an AC to DC converters (wall wart)?
Wall warts are designed to meet a certain balance of cost, size, efficiency, reliability, output quality, and output power.  Each wall wart will have a different balance.  Apple might focus on size and efficiency, while Samsung might care more about cost.  The designer of the wall wart will understand the goals, the balance, that is required and design the wall wart accordingly.

There are too many designs to really go into the details about what makes one design have higher output power than another.  Suffice it to say that EVERYTHING matters.  Even the plastic case that the wall wart is inside matters.  Just like if you are designing a passenger airplane to go the farthest distance, then even the seats matter.  For a wall wart, the transformers, caps, diodes, wires, integrated circuits, inductors, resistors, MOSFETs, all matter.  You can't just change the caps and expect the whole thing to suddenly output more power.  You have to look at everything.

Let's use a car as an example.  Let's say that you have a typical car that you want to make go faster.  Of course you could replace the engine block to hold more and larger pistons.  But then you need a bigger air intake so you can burn more fuel.  And you need a bigger fuel pump.  Now you need a stronger crankshaft, to handle the power.  And a better transmission, because the old one would get destroyed under the stress.  And a better driveshaft.  And better axles.  And better wheels/tires.  And better brakes, so you can stop this.  And better suspension to handle the increased torque.  And better radiator to cool the hotter engine.  Etc.  The same is true for a wall wart-- but with electronic stuff instead of gear-head stuff.

Everything matters.