> it would still be like blindly throwing a dart and hitting a bullseye 24,000 light years away in this case.

I liked this analogy, so I felt inspired to calculated the odds. Particularly: the percentage of the "sky" (celestial sphere) that is covered by the Earth from Sagittarius A* 's perspective. That's equal to the two dimensional projection of the Earth onto a 26000 ly sphere around Sagittarius A*.

  cover% = π x (6371km)^2 / (4 x π x (26000 x 9.46 x 10^12 km)^2) x 100%
  = 1.677 x 10^-26 % 

I tried to put that number into "human terms", but the best I could come up with is it's the same as the width of a hair at Saturn's orbit, viewed from Earth.

Slightly off topic, but one of my favorite solar system facts is that the with of Saturn's rings can be measured in feet. They are much thinner than I ever expected.

Sure, if you consider 3,200 feet as "measured in feet" [1]. Anything can technically be measured in feet. When you said that I assumed you meant they were like 20-30 feet thick, not over half a mile.

[1] http://coolcosmos.ipac.caltech.edu/ask/108-How-large-are-Sat....

From a NASA website[0]: Saturn’s rings are incredibly thin. The main rings are generally only about 30 feet (10 meters) thick, though parts of the main and other rings can be more than a mile, or several kilometers, thick.

[0] https://hubblesite.org/quick-facts/all-quick-facts

That's still pretty damn thin, relative to the size of the rings. An interesting quote from your link:

"If you had a model of Saturn that was a meter stick wide (3 feet), its rings would be about 10,000 times thinner than a razor blade!"