No. To both questions. It's not legal to use software to circumvent Youtube in order to download and make copies of songs or videos. That's copyright infringement. Because of this, there is no underlying legal use for that sort of software - it's contributing to copyright infringement, probably willfully. There's no difference between what these sites are doing and what Napster and BearShare and those sites were doing back in the day. It is illegal to make copies of songs and movies from the internet. End of story.
No information you obtain from this answer is legal advice, nor is it intended to be. You should consult an attorney for individualized advice regarding your situation. No attorney-client relationship is formed by my responding to your question.
Q:1.) Can an individual in the United States access the software being hosted on these websites and use it to download a copy of a copyrighted song hosted on YouTube, without infringing on this copyright and if the song is only for personal use (and if the song is not further distributed)?
A: NO, that is copyright infringement. The download is illegal unless you get permission and what use you make of it after you download does not change the infringing nature of the download. Also, there is not personal use exception that allows you to make copies of infringing copies. If the original copy is infringing, so is s a backup.
Q:" A relevant analogy in which people have taken a private copy of a public broadcast would be people recording a radio program with a cassette recorder or making a copy of their favorite TV movie by using a video recorder. "
A: No that's not relevant and not analagous because a limited broadcast like YouTube (read the TOS you clicked as agreed) is not an unlimited public broadcast.
Q:"2. Is it legal for a company based in the US to provide these YouTube to Mp3 copying services such as the ones being provided on the aforementioned websites, especially if the company only intends these services to be used in the making of legal copies?
A: NO, and that "intention" is pure BS anyway. Napster, KaZaa, Grokster, MegaUpload and Limewire all tried that lie and it did not work.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.