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I use Opera and the "All2MP3" utility to do what you're looking to do.

My workflow:
1. Opera with "YouTube Downloader Lite" extension installed.
2. Go to music file on YouTube
3. Download it as an "mp4" file -- highest bit rate.
4. Launch "All2MP3" (app)
5. Drag-and-drop mp4 file into All2MP3.
6. Choose highest bitrate and choose option to delete original file after conversion
7. Let All2MP3 do its thing
8. Done.

There are "online converters" available, but I like my workflow above because it gives me control over what is downloaded and over the complete conversion process.
I'll often do this "an album at a time"...
 
I use Audio Hijack from Rogue Amoeba for things like this. I am also a frequent user of their Airfoil.

Audio Hijack "hijacks" or records any audio from any app to if not any, at least many formats.

Humbly, Ylan
 
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