HC-06 RX pin usefulness
If you never want to change the configuration of the module either, then no, you don't need to connect the module's RX pin.

However many people want bi-directional communication. Not only the ability to send a signal to turn the LED on or off, but also the ability to get the current state of the LED - is it on or off?

Or maybe to send an analog reading. Or the data from some sensor.

The HC-06 is a bi-directional module. If you choose to use only one direction then you only need to wire in one direction. However, tie the RX pin to 3.3V with a 10KΩ resistor to stop it floating and consuming extra power or causing random noise to be transmitted to your phone.

In fact, it's just like a phone. You can call someone up and just speak to them without listening to what they are saying. That's fine if all you want to do is call Mom and say "Pick me up in 5 minutes" and hang up. But if you want to ask your Mom "Can you pick me up in 5 minutes?" you have to actually listen for her answer "No, walk home you lazy oik!".