I found some answers for what I was wondering about! This scholar, Ash Branch, shows that Tolkien was shy about aligning astronomical events (such as moon phases) in his stories with historical ones. That was part of why Tolkien avoided specifying whether it happened 6000 or 8000 years ago.
With that much flexibility, you *can* use astronomical events to nail it down to a date that works. Branch reckons that in "The Lord of the Rings," the Fourth Age began in 3102 BCE. Then other specific dates follow from that, including the reasonable conjecture that Tolkien might have felt that World War II was the end of the Sixth Age.
I'm only starting to explore the rest of Branch's site since I found it today, but it looks like the site is meant to be about an earnest Tolkien-inspired Pagan spirituality.
In any case, I see that makes our current year, 2021 CE, the 67th year of the Seventh Age.
The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!