Aulani can only be as crowded as the maximum capacity of the resort allows. So while the pool and such can feel more crowded than other times, its not as drastically noticeable as a peak crowd at WDW would be (for example). That said, Aulani is notorious for not having enough space and amenities to accommodate a resort booked to capacity. This has been said its because the developers anticipated people using Aulani as a "home base" for Oahu touring, and underestimated the amount of people interested in staying put at the resort without leaving. Because of this, Aulani at the busiest times might be worse than a comparable resort (i.e. Marriott Ko Olina) at the busiest times. I mention this as consideration for doing a split stay with another resort, or perhaps choosing another resort in Ko Olina where you can still see and dine at Aulani, but have space and more amenities. In the current environment with so much at Aulani closed and no character meets, there's fewer things that actually set it apart from comparable resorts.
I agree with PP that if you can visit during February break, if you have that, that's the best time since it seems to be a Northeast thing and fewer people are off. Winter months are peak season for Hawaii as its prime whale watch season and people want to escape the cold on the mainland. Another option, depending on your school schedule, is late August/early September. Schools here start back after Labor Day, so there's a typically a couple weeks in that timeframe where many schools across the country are back in session but many of ours here in the Northeast have not gone back yet.
November (Veterans Day or Thanksgiving) is probably a good time too. I would avoid October Fall breaks/Columbus Day, as that is a time Hawaiian schools have fall breaks from what I have seen.
First, there are character meets all day long at Aulani (distanced) and I have never had such short wait times (if any) to meet characters -- it was fantastic! Each day you dial "CHIP" on the hotel phone and they give you the greeting schedule for the day. My kids loved it and we got a lot of great character photos without waiting.
Second, my personal opinion differs from yours on Aulani being set apart from other resorts. On Oahu, I think Aulani is in a category of its own. The theming, beauty, incredible service, Rainbow Reef, number of pools/two slides/lazy river -- I don't know of another resort on the island that compares for families. Also, in normal times, they have the kids club included which is a vacation-making amenity for parents (and kids) and I don't know of any other resort that offers this type of amenity as an inclusion in your stay. Also, I have heard the spa at Aulani is incredible although I have not visited myself as it was closed when I was there recently.
I have not visited the Marriott Ko Olina, so it might be equally incredible and I just don't know about it! I do know that right now, you cannot spend time at the pool without a reservation, and you are limited to one 2-hour reservation per day. This may change if/when distancing requirements are relaxed.
Last, just FYI right now, you cannot dine at Makahiki or Off the Hook (the only places you can sit at a table) at Aulani if you are not an overnight guest there.
Right now, Aulani is maxed out at 65% capacity which is really not too bad (other than the elevator lines and the need to claim pool chairs as the sun comes up). Again, this will change as distancing requirements are relaxed but if the OP visits this year, they can have some peace of mind knowing it won't be insanely crowded.
I will also add that when I was there President's Week, there were a LOT of local families that come in on the weekend using great local discount rates, so from Thursday through Saturday nights, it was crowded (maxed out/sold out at 65%) irrespective of mainland guests.
