I don't think it would be easy to find the origin of liver flush, without spending a lot of time and energy ...
Olives & Olive oil are traditional foods of people living close to the coast of Mediterranean Sea.
During the last 35 years of my life, I spent from 1 week to 1 year at the coast of Mediterranean/Adriatic sea.
This is the information from the book: "Curing with Herbs" by Dr. Jovan Tucakov:
"Olive oil is used as a mild remedy for gallstones, colic and constipation. In large doses, olive oil is used as Antidote against toxins that cause irritation. It calms sun burns. No other oil is as often used in pharmacology as Olive Oil.
For constipation, take 1 -2 tbs in the morning.
For Gallstones expulsion, take 2 tbs before sleeping, 3 - 4 days in a raw."
The Master Herbalist Jova Mijatovich, in his book "Herbs and remedies" list olive oil as #1 remedy for liver pain, and #1 remedy for gallstones. Beside olive oil, he lists 12 other herbal remedies.
His book contains many hundreds of traditional herbal remedies for every possible ailment.
Recipe for liver pain: Before breakfast, on an empty stomach, take 1 tbs of olive oil, and then one tbs of Lemon Juice. He doesn't say for how long time to do it, but we know that it is safe to do it for weeks.
For gallstones, he gives similar recipe:
"Before breakfast, on an empty stomach, take 1 tbs of olive oil, and then one tbs of Lemon Juice sweetened with raw honey. Drink it every morning the first week, every second day the second week, and 1 morning the third week."
I have spoken with a friend from Morocco, and he told me that olive oil flush (larger amount of olive oil) is a traditional remedy in his country. He himself has not tried it.
The recipe my grandma Nada got around 1950, in Belgrade, must have come from the coast of the Adriatic sea, cause it involved Olive oil.
In many places along the coast of Adriatic sea, Christian monks and other people working in monasteries are the main producers of olive oil. I have spoken with Christian Monks from coastal monasteries in Croatia & Montenegro. I always question them about traditions, traditional remedies, and whenever person suffers gallstones, they recommend 2 tbsp of olive oil on an empty stomach, the first ting in the morning, with lemon juice, and they say it works.
Some of them were aware of the recipe involving 1/2 cup of Olive oil, but they have not had personal experience with that remedy and they were not recommending it.
I have also spoken with olive oil producers in France & Italy, and most of them never heard about the liver flush, but all of them were aware of the 1-2 tbsp of oil in the morning, if you have gallstones.
Olive oil is always used with lemon juice.
Have this recipe originated at one place, by one person, and then the information was spread by the mouth, or is there a possibility that on many different places people experimented with olive oil for gallstones, and after increasing the amount of oil, they discovered the flush?
Both options seem possible, but considering that people have used olive oil for many thousands of years, and considering that even Egyptian Pharaohs suffered from Gallstones several thousands of years ago, it is possible that olive oil recipe is older then we think.
One of the first books in USA that published Liver Flush recipe was "One Answer to Cancer" by Dr. Kelley.