There is no problem with the Voltage Regulator.
On the Arduino Uno board, the voltage supply is flexible. Either +5V on the +5V pin, or an adaptor to the voltage regulator, or USB power.
The onboard voltage regulator could get hot. And the USB power is often less then 5.00V and only 500mA. If you need more current from the 5V for your project, and external +5V power supply is needed.
A zener is not a voltage regulator.