I’m a DVC member who owns at Aulani and normally stays there every 2-3 years. When I stay at Aulani, I typically stay 7 nights and I always rent a car from the Alamo (reserve in advance) on site for the days I plan to leave the property. As OP have said, unless you plan to eat on property or across the street, you will need a car while there to sightsee and get food.
Most times I have visited Aulani, I normally wind up spending 2-3 days driving to the Waikiki area for activities. While I’ve always wanted to stay in Waikiki to cut down on some of the driving, I finally decided to do so for my visit this year. Instead of staying at Aulani, I was able to get a RCI weekly reservation at The Imperial Hawaii. I’m super excited and don’t plan to rent a car while in Waikiki. If you’re ok paying OOP, you can go to their website and check if they have availability for the days you want. I also researched the following properties while I was deciding what to do: Outrigger Waikiki Beachcomber, Outrigger Waikiki Beach Resort, Hokulani Waikiki by HGV, and
Surfjack Hotel and Swim Club. If you have additional points you can use, you can also see if you can get a RCI reservation for 2 days at a particular resort. If you’re eligible, there are also two hotel properties in Waikiki that are available under the Concierge Collection which you can use your points to book.
While Aulani has tours that leave from the resort, most independent tours leave from Waikiki and you will have to drive there to get the tour. Unless you plan to tour on your own, this may play into your decision on whether to stay in Waikiki. Aulani is a destination in itself and a great place to stay, especially when using DVC points. However, your experience staying there will be very different from staying in Waikiki which is somewhere that is very walkable and has a plethora of restaurants in every price range.
IMO, if you want to stay in Waikiki do so either before or after staying at Aulani. Before, if you want to get to your resort quicker and plan on doing things on that side of the island. After, if you want to be closer to the airport when you leave and plan on doing tours at the end of your trip. Keep in mind that there aren’t many properties in Waikiki that rival Aulani. Good luck with your decision.