Why are colours SO MUCH duller if you change the Mode to CMYK ?
I know, I know, CMYK colour space is smaller than sRGB so some colours look dim/dull yada yada yada. I'm well aware of that, but that's not the issue I'm facing.
So I'm working in a Photoshop file and forgot to set CMYK colour space at the beginning. When I go to Image > Mode and set it to CMYK the colours looks CRAZY dull, almost as if there's a white filter with 50% opacity set over the whole image. It doesn't make sense. I have experience in printing and I've seen what printers can do, but my orange colour I'm seeing looks more like gray than orange. When I go to websites that convert HEX to the closest CMYK colour my orange looks SO MUCH more vibrant than whatever I'm seeing in Photoshop.
But, if I don't do that, and instead go to Edit > Convert to Profile and set CMYK there than the image looks vibrant. And if I go to create a new file with CMYK at the start, upload my exported png there that the colours also look vibrant, yet it's CMYK mode from the start.
What is happening? Shouldn't Image > Mode be the right way to convert colour profile? Why then there is such a huge difference in colors when I do that compared if I convert the profile in Edit > Convert to Profile or create a new file with CMYK already set? It makes zero sense!

