I still love Avenue Q because I was a teenager when it came out and I know the context for a lot of jokes that have aged worse than milk. Once upon a time Christmas Eve wasn't the butt of a racist Japanese joke, the joke was about weebs and she was making fun of them (perhaps best seen in the line "tried to work in Korean deli, but I am Japanese"). In 2003, a gay Republican would have been seen as silly and a bit deluded but not dangerously ignorant because it was a very different party back then. "Everyone's A Little Bit Racist" is...okay it was always a little sus, but I think if you can take off your Earbuds of Self-Righteousness for just thirty seconds and listen to the words being said, you'll realize it can be taken as a serious warning about unexamined biases. (It likely wasn't intended that way. That doesn't mean it doesn't work.)
So: Hamilton? Hamilton came out at a time when the US was having a reckoning with race and immigration history. If you look at it in the context in which it was created, it makes a lot of sense. Yes, it's clumsy and flawed. Most early attempts at any kind of social grappling are. But it was trying to ask us to look at something we very much took for granted (of course everyone in the story of the Revolution was white...even though that's not actually true, and in fact the first death of the Revolution was a Black man killed by British soldiers at the Boston Massacre), and to really think about it, and to imagine a world that was different.
And to add to that, even it being flawed doesn't mean it did no good. It did make a lot of people think! It forced a lot of people who gravitate to musicals because of a very Eurocentric musical style to stop and think about traditionally Black genres like rap and R&B as not just music, but art, and to think about the implicit and explicit biases in musical casting and production! Up until that point, the vast majority of Black-specific roles were "addict," "slave," "cautionary tale," or "joke." Suddenly we had a whole slew of them available that were "dignified, honorable, petty, messy, passionate, complicated people, sources of inspiration and foundation." I love musicals, but I'm not going to sit here and pretend Broadway doesn't have a racism problem, or that it wasn't even worse before Hamilton. I can name exactly two pre-2016 musicals that feature Black-specific roles that aren't the above-named archetypes: RENT (which has one: Joanne, with Benny being a cautionary tale about forgetting your roots) and Hairspray. And while Hairspray is great, it's also "slaves 2.0: Civil Rights Movement Edition," and a white girl saves the day. As I have bemoaned before: "are Black people allowed to tell one single story about themselves that isn't about their oppression?"