There's more than one way to skin a cat, so they say. And there's more than one way to get monochrome from a color image. Since YUV is a common encoding method, the easy thing to do is to throw away the U and V, and just keep the luminance, effectively doing the same thing old black and white televisions did.
But in this modern, digital era, it's possible to control the balance of each color channel's contribution to final black and white image, and to do it with free software.
In the free version of DaVinci Resolve, you can do this in the RGB mixer panel of the Color page, by checking the "Monochrome" box, and the "Preserve Luminance" box, and then moving the R, G, and B sliders to taste. You'll notice that certain combinations give better contrast to skin tones, among other things, and that small changes can dramatically influence the resulting image. It's also a great exercise in learning how to expose a camera well, and learning how camera exposure and color are related.