I was talking about this with a director colleague of mine. For a long time, I admit I had issues with some aspects of cross-racial or color-blind casting -- especially if actors of different races were playing members of the same family. I eventually got over it and no longer care, and actually look forward to it now.
My own feeling is that, if the issues raised in the play/movie/television series aren't dealing with specifically racial issues (as "Othello" does, among other things), there's no reason to not cast anyone in any role. I wouldn't cast white people in an August Wilson play than I'd cast a Black actor in a David Henry Hwang play that was about the Chinese experience.
About a decade ago, I saw a production of "West Side Story" that had a
black Tony. I have no doubt the director was sincere in casting
color-blind, but it made no sense in that there was no mention of it.
The racial conflict at the heart of the play was rendered nonsensical.
(It didn't help that the actor wasn't very good.)
I'd certainly use actors of color in Shaw, Chekhov, or Wilde plays, though, because those plays aren't about race. (Though it's got to be done carefully. There was a terrible production of "The Cherry Orchard" on Broadway a few years ago that used a multi-racial cast, but -- whether through design or not -- used Black actors in the lower-status roles. It cast an unpleasant pall over the proceedings [although most of the cast was floundering anyway]. Would I have had any objection to a white Ranyevskaya and a Black Gaev [her brother]? Not at all.)
In my own case, I've cast a Chinese-Canadian actor in a role written for a Latinx actor (we didn't get any of the latter at the audition) because what the character was going through was more about an immigrant experience (running a family-owned restaurant) than it was specifically Latinx. (I also used the same actress in another production playing a couple of characters written as white men. There was no reason the characters had to be white or played by men, so in she went). I've also had Asian actors playing main characters in "Guys and Dolls," a production that also featured a number of women playing gangsters. I did a series of readings of my translations of Chekhov plays last year and cast all sorts of people from different backgrounds.
"Hamilton" has a few sins -- and is already seeming a little dated -- but what it proved to me more than anything else is that, in almost every case, any actor can play anything. This Lorre thing is one of the exceptions in that the character is specifically Afghan and dealing with the Afghan experience, and should be played by, yes, an Afghan. I wouldn't cast an Indian in the role any more than I'd cast an English, Nigerian, Icelandic, or Singaporean actor.
As for your specific examples, the racial issues in "Othello" can be substituted for issues about "others," so the actor playing the role should probably be another ethnicity than the rest of the cast. (I have seen at least one production where both Othello and Iago were Black, which made it more about jealousy and less about race than usual.) The Patrick Stewart production you cite seemed like more of a novelty than anything else.
I could barely get through one episode of "The Great," but not because of the casting; I couldn't stand the tone and writing. It was great to see actors of color get those opportunities. If I could abide "Bridgerton," I'd probably feel the same way.
And don't even get me started on Brits and Australians being cast as Americans.
--Dave Sikula