A web search says your approach (which I admit I don't follow) is off by a factor of 40 -- the Sun moves approximately 500k mph around the galactic center. A speed 8x slower than the star mentioned in the article.
The GP is talking about the sun’s apparent speed in Earth’s sky, which is dominated by Earth’s rotation. Another way of stating it:
The sun is 1 AU distant and appears to revolve around Earth once a day from our perspective so it has apparent speed of 6.28 AU / day or 2.4 million mph.
Their point is that unless this star was very close to us it wouldn’t appear to move quickly across the sky, because the sun has similar apparent velocity and looks stationary to our eyes.
"Quick" being relative to your expectations, of course.
If OP is right, and it moves six times faster than the sun, then it would go from one side of the sky to the other in just three nights, right? That's extremely fast for any star gazer. Faster than any planet we can see. Someone who was familiar with the night sky and who could see this star would notice its speed quickly.
