Is it correct to say that a smartphone cable is only a "1A" cable?
What happens is the thinner cables, having a higher resistance, will drop V=IR volts across the cable run. Which in turn causes the phones charging regulator to drop down to another standard charging level when it sees the voltage at the phone end be much lower than it expects. While USB is supposed to be 5V, phone chargers have been made to be 5.3V or higher on the charger side to make up for the voltage droop across the cable at higher amperage, and newer standards are using higher voltage at lower currents to sidestep these issues as well.

While he's painting the issue with a wide brush (it can only do x amps), in practice the actual current drawn will depend on the specific resistance vs length vs current the specific phone will draw, so he is correct. Cables with thicker wires will allow a higher charge.