Some people handle fats better than others. It all depends on the individual circumstances.
Sometimes upping fat intake, being mindful of what types of fats the individual does best on (ie raw fats, saturated, high omega 3, etc etc) can just provide enough bile stimulation to clear things out. Fats are really weak acids, hence the name fatty acids. As such they can actually break down cholesterol stones inside the liver, but for this usually a person must go on a strict diet and limit Sugar and carbohydrate intake.
Stone formation and liver congestion is associated with a low fat diet for some people. They then retreat, eat low fat, feel somewhat better because they're not making things worse and then assume low fat is the way to go. This is not good in the long run. In general I'd say if you're kapha constitution you lean toward low fat, light and dry diets, pitta lean toward cooling moderate fat diets and vata usually does well with higher fat/oily diets.
coffee enemas are one way to cleanse the liver (and colon) but by no means the only way. There are different strategies to use depending on the individual. Some people do well with bitters, which stimulate bile flow. Some people do better with liver flushes, some with coffee enemas. The herb phyllanthus amarus, incorrectly known as chanca piedra sometimes helps break up liver congestion.
Iodine can help to raise body temperature and metabolic rate, thereby in some cases aiding the breakdown or melting of cholesterol stones. The liver is very temperature sensitive.
Coconut is actually a good fat to *try* at least since mct oils do not require the liver for processing. That is probably why whoever recommended coconut did so - it was a good idea imo. But you always have to gauge your own responses to things because rarely does anything work out like internet people say.
Lecithin helps keep bile soluble and emulsify fats when bile is being created. A good natural source of lecithin is raw egg yolks. Some people are sensitive to the type or source of lecithin. You can try raw egg yolks, sunflower or soy derived lecithin, or egg yolk derived lecithin powders.
Some people do well with supplemental methionine and or NAC.
This is a good discussion and very important information to keep in mind when any 'detox' is embarked on. If your liver is not releasing toxins you are not really detoxing.