Is it really OK to supply more current than what the component is rated for?
Your misconception stems from one incorrect statement: Since most power supplies provide a current that is higher than what most LEDs can handle, you must put a resistor in front of the LED in order to not burn it out.  

The reason for the resistor in series with your LED is that if your power supply supplies a higher voltage than the LED requires, and your power supply is capable of supplying more current than the LED can handle, then you must limit the current your circuit draws from the power supply by using a suitable series resistor.  

A 5A power supply will not force 5A through anything you connect to it. It will only allow up to a maximum of 5A to flow and how much actually flows depends on the power supply voltage and the effective total resistance of the circuit connected to it.