This is an interesting puzzle and I tend to obsess on things once I start thinking about them, so I continue to think about all this. Kava is a bit unusual in that it is an emulsion of something that is not soluble in water, but seems to emulsify more easily than lipids. Water is a strongly polar protic solvent. Ethanol and acetone are somewhat less polar and aprotic. Water forms very strong hydrogen bonds, as evidenced by its high surface tension, and one of the reasons that it makes good emulsions. So, high surface tension is one of the properties that you would be looking for if you were looking for an alternative solvent. An extraction with an alcohol is not an emulsion, in that case you are using a solvent to actually dissolve substances, and it is suspected that is where you get into trouble with solvent extractions from kava, substances like kawain may dissolve well in alcohols and ketones, but for some reason do not emulsify as well as the desirable kavalactones. It has sort of the feel as a liquid-liquid extraction where you are trying to separate things that are differentially soluble in two different phases.
An interesting experiment would be liquid chromatography with a solvent gradient. I am mostly familiar with using ionic strength or pH gradient, but I think it could be done. Probably what you would do is load a column by suspending solvent-extracted kava in a neutral matrix with water. Then you would set up a gradient with pure water and alcohol or acetone. You would take fractions of several ml and later these would be analyzed ideally by gas chromatography, trying to get some idea of what concentration the good stuff comes off and when the bad stuff comes off the column. Ideally, you want to get good separation of the bad and good substances, assuming that is even possible using a solvent rather than an emulsion.
Another problem with kava is that if you were going to try something like a soxhlet, you need a solvent that boils well below about 60 C at a temperature that the kavalactones are stable. Alternatively, you could hydrolize the lactone turn it into a soluble salt, crystallize, redissolve and form the lactone again, but that will make a racemic mixture depending which side of the carbon forms the lactone, and I don't know if anybody knows the properties of the other enantiomer as it probably does not occur in nature as enzymatic reactions tend to be stereospecific. Possibly you could co-crystallize with pure R or L carvone and that might separate the enantiomers fairly well, but then you would need to determine which enantiomer is crystallized and if it's the one that you want. This is probably more expensive and a bigger PITA than justified if you are not making this in bulk.
A Deep Eutectic Solvent (DES) could be interesting especially glycerol with choline chloride, which is used to remove glycerol in the production of biodiesel. I'm not sure what temperature it boils at but know that the melting point is much lower than either water or glycerin, so it might be low enough. I still think this is a bad idea, but damn it's an interesting chemical engineering problem. I think supercritical CO2 is probably the best way and kava extract produced that way is available from Paradise, but still no guarantee that the product is completely safe, and obviously it requires specialized and very expensive equipment that is hard to justify if you're not making it in bulk.
I also thought some about how other emulsions are produced in general. I think that shaking vigorously, like in a closed jar, could be interesting and possibly work better than a blender, I think that one might be worth trying. Mortar and pestle could also be interesting.