In the 1960s we were experiencing multiculturalism for the very first time. The Vietnam war had our troops fighting alongside Asians and troops came from around the world to fight alongside the US. When Gene Rodenberry envisioned the series, it was his idea of the old TV series Wagon Train, set in space, with a cast exploring new frontiers. Remember that line from the show into? "To explore new frontiers". Back then the US and Russians were the only ones in space, and we were both striving to reach the moon. He envisioned a world effort, like he saw in World War II. Think of the cast . Shatner and Doohan were Canadian, and one of the few white Americans was Learnard Nimoy and his parents were Russian Jews who emigrated to the US and settled in Boston. Only Deforest Kelly was a fairly well-known actor from his previous roles in westerns.
That is why Blacks, Asians, and non-Americans were cast. Look at the crew of the International Space Station today. Do you not see that there? People from 23 countries have visited the ISS. The Chinese have their own space station now. Rodenberry predicted the future very well. I actually met him when I was in college at my university where he was a guest speaker, and talked about some of these traits.
Producers didn't accept women in a leadership role in Star Trek until much later. In the pilot, Majel Barett played Number One, Captain Pike's second in command. Producers were having none of that and the pilot was redone, and she was recast as Nurse Chapel. Had she stayed, that would have definitely qualified as woke. Star Trek: Strange New Worlds returned to a female Number One, just like the pilot, now played by Rebecca Romijn. Captain Pike, and Spock still remain from the original pilot that was incorporated in the original series episode Menagerie parts I and II. Captain Pike, Number One and Spock's actors and characters all appeared in Star Trek: Discovery also.
Were the cast and characters diverse? Definitely! Was the cast woke? Not really. The original series addressed many social issues. If talking about them was woke, I guess you could say it was.
Diversity yes, but woke? Not to me, but your mileage may vary.
Yes, I have been a Trekkie since I was a kid and that's about as long as it has been around.
Live long and prosper.