You asked, "How is it that a free to listen to YouTube recording of an old album being recorded for personal use is somehow an infringement on anyone's bottom line?"
It isn't... but since money is at stake, come hell or high water these companies will do everything they can to make sure you can never legally make a copy without paying top dollar for the privilege.
Technically the law says nothing about money and it's only illegal to make a copy without "license" (permission). But naturally, if you want a license, the typical arrangement is you pay for a copy from an approved vendor, and maybe a few pennies will make their way to the artist's royalty account, with the rest going to pay for the middle man's hookers and blow.
The reason music is freely available to stream from YouTube is because the DMCA gives YouTube safe harbor from liability for what its users do, so long as YouTube jumps through certain hoops. The users are still on the hook for uploads, at least, but it is usually not worth the expense and bad PR to sue them, so people upload lots of stuff without permission.
Very few countries have a blanket personal-use exemption that allows just any noncommercial copying. Most countries do have fair-use clauses which have led to court cases where copying by consumers was determined or implied to be OK, but these are limited in scope and rarely apply to wanton copying of entire works, especially if the alternative is you just buying a copy.
Besides, think about it: when you buy a copy, that's just for your personal use, too... so how could personal use be the thing that makes you entitled to get it for free?
Nevertheless, it is a weird and absurd situation where you can stream YouTube vids all the live-long with no consequence, yet if you dare save one of those files to listen to later, you're a lawbreaker.
Anyway, to cover their butts, YouTube has made it so that you can't get a video file without using their app or their player in your web browser, and on your computer the file stays in a secret location and is discarded as soon as possible, and they don't give you a save-to-somewhere-else option. Nowadays it's also sometimes downloaded in pieces so you never have the whole thing at once.
Knowing where to look or using a browser extension to grab the video file out of the cache before it disappears used to be an option, but generally isn't anymore. Intercepting the download and saving it to a file you can keep is still possible with illegal stream-capture software. There is also software which is semi-legal and just pretends to be a browser, but gives you a save-to-file interface. Such software is generally for power users, and is not something that would meet your criteria for being easy.
The only really easy option is using someone else's illegal website which acts as a middleman, pretending to be a browser and fetching the video file and sending a copy to you, perhaps with some additional demuxing (separation of audio from video) and transcoding (conversion to your preferred format like MP3). The most popular of these sites got shut down recently, but there are many more out there; a simple web search for youtube download will surely help you find them.