As you have noticed, just changing the file on disk isn't enough. You have to tell MC that you are changing cover art. Which is actually very easy from standard view (the normal computer interface). I don't think you can do this from Theater View, but I'm not 100% sure.
From Standard View, just right click on the album cover (or all of the files in the album, highlighted at once) then Cover Art > Add from File
or
Cover Art > Paste from clipboard
I usually use the Paste from Clipboard option because that means I don't need to download a file. I just find what I want with an image search, then copy the URL to the picture, and paste it into MC. MC takes care of downloading the file, making it Folder.jpg, and embedding it in the file.
Speaking of, you used the term embedding. Embedding puts the image *inside* the FLAC file. So Folder.jpg file on disk, is unnecessary. Because every FLAC of that album will have the album art inside it. This is my prefered method because it means the art can never become separated from the music files.
MC has an option to do this. I'm not sure if it's on by default. Find it here:
Tools > Options > File Location > Covert Art > also store image in the file's tag > (checked)
I hope this helps.
Brian.