The oldest living tree has seen 9,565 birthdays (https://en.wikipedia.org/wiki/Old_Tjikko), but it doesn't have 9,565 rings. Even if it did, there's a gap of at least a few thousand years ;-)

I'm impressed they got it exact, but wasn't by ring-counting (only).

Not necessarily. You can compare dead trees in the same vicinity that were alive at overlapping time intervals. Patterns in the width of the rings will reveal the period over which their lives overlapped. With enough dead trees, if they are preserved well enough, you could go back significantly further and age trees that are long dead.

It's called "Dendrochronology" and this is exactly how it is done. In the Northern Hemisphere they can date back at least 13,910 years.

Trees turn into fossils too sometimes.

You still have to figure out when it did. If there's a tree with 800 rings, but it died 100,000 years ago, then there's 100,000 years you have to date by some other method, and add 800 on top.

That’s not how they got to that figure, which is 50% greater than the oldest known tree… (and I don’t think they even counted the rings for that one)