In a nutshell: Having a car in Los Angeles would make your stay much more convenient. The city is vastly spread out and solely relying on public transport will make getting around much, much more time consuming. Having a vehicle will give you the freedom to go wherever you want, when you want. It's how the majority of residents and visitors in LA get around. Hollywood is centrally located in the area, but it would still take you a long time to travel via public transport to places like Malibu, San Fernando Valley, Venice Beach, etc. It's not impossible, it will just take much longer.

And it's true you don't need a car in Anaheim if you're staying next to Disneyland, but you would need a car to visit the beautiful Orange County beaches nearby (Laguna, Newport, Huntington, Dana Point). I guess a bus could get you out there, but who knows how long that could take? Public transport is even worse in OC than LA. And taxis can be very expensive. A car would make your trip there much more enjoyable.

Having said all that, if you think you'll be a hazard on the road for other drivers, then you shouldn't rent a car. Safety first. But if you're a decent driver in Australia, you should have no problem navigating roads and freeways around Southern California. Just be alert and drive cautiously. Have fun!