No, you almost never want to embed the pck, unless for very small simple apps. When you make a game, usually you would submit it to a store like Steam or Itch, in which case it needs to have a separate pck in order to release patches. This is because exe files cannot be diffed, meaning if you add a 1MB image to a 500MB game, users will have to download 500MB instead of 1MB. And if you use Steam, for example, you can submit your own icons in addition to the ones in the export settings, and you don't need to do anything with hacking the icon.
If you are distributing yourself, I would still recommend leaving the exe and pck separate (you can put them in a zip to download, along with a readme, license information, etc.). You can use Resource Hacker to change the exe icon and I've used it and I know it works. It may or may not work with embedded pck, I never tried it due to the warning in the documentation. But it does work with individual exe and pck.