Fantastic find!

That YouTube video is blocked in the USA (and only the USA interestingly). I have to VPN to Canada to play it. But the link works!

So, final details:
- The piece is at the very tail end of the comedy piece "The New Technique Psychiatric Society".
- The piece is found on the album "2000 & One Years" by Carl Reiner and Mel Brooks.
- Natalie personally liked the piece and used it in the film.
- Natalie's sister, Lana Wood, wrote about it in "Natalie: A Memoir by Her Sister", but got the title slightly wrong when she wrote about it.
- Link to the piece on YouTube: https://www.youtube.com/watch?v=lUaTBHECJhU&t=444s
- The YouTube copy of the piece is region-blocked in the USA, but can be played by US residents by VPN'ing to another country.

Thanks so much, everyone! Mystery completely solved.