How do I design a 2A or more power supply for my consumer USB devices?
The reason why your Apple/Samsung devices do not draw more current is simple.
It is because there is additional data communication going on between the Apple/Samsung device and their dedicated power supplies. This makes both devices recognise eachother and agree on a higher current to be used. The charger sets certain voltages on the USB data lines and this is recognised by the phone or tablet.

Your 20 Amps supply does not supply these voltages on the date lines so it does not "talk" to your Apple/Samsung device therefore these assume it is a normal "dumb" charger and do not draw more current than what is allowed by USB standard which usually is only 100 or 500 mA

To charge an apple device put thoses voltages on the data lines: 

desired current : 
2,000mA     D- : 2.0V D+ : 2.75V



Also, depending on how full the battery is the charge current is also limited. It will only be maximum when the battery is 30 - 70 % charged (these numbers are just my guess). Charging with a high current is bad for the battery when it is very low or almost full.

Sources: 

Adafruit : The mysteries of Apple device charging

Voltaic : Choosing USB Pin Voltages