Remove background doesn't always work

With a graphic image like this it seems sensible to create Solid Color Layer based on the Channels that offer good separation anyway. 

If this is for print it might, provided the effective resolution is high enough, be useful to split it into three bitmap images and assign those the respective colors in the layout application (indesign, Illustrator, …). 

That's true.  Automatic tools are not always as precise as manual selection methods.  A lot depends on the image you're working with and the contrasts between foreground & background elements.

I like the temporary Quick Mask Mode in Photoshop.  It allows you to refine selections in an image using a semi-transparent overlay. It's a useful tool for making precise selections around complex shapes.

https://helpx.adobe.com/photoshop/using/create-temporary-quick-mask.html