In general I think we're on the same page. You definitely want your liver, kidneys, and lymphatic system open and running clean. How to get there, well I think we'd probably disagree on the particulars.
I also agree that there are some people who get into BIG trouble with iodine. Mainly due to the T component of VWT.
I have used high raw fats to cleanse the intestines of plaque so I can say for a fact that at least in 1 case (and there are a lot of other people who were doing and are doing raw paleo that can testify to the same). Your body/liver might not do well with certain fats, that's fine, but not everybody's metabolism works the same. At a certain point years later I switched over to mainly fruits and carbs, so again at least for n=1 I think it's a dynamic situation.
As far as I last researched the oxidized cholesterol plaques were mainly an arterial problem. It could very well be a cellular problem too, which is why raw fats are safer. If you consume raw vegetable fats then you must extract them yourself to ensure they aren't rancid. And even in the case of good quality flax oil it can oxidize or become unstable as soon as you ingest it (see Budwig for more). There are a lot of twists and turns and assumptions as to what is going on in the entire processing and digestion chain.
BTW some people get into even bigger problems with lecithin than they do with iodine. It is such a good arterial scrubber that large pieces of plaque can get dislodged, hit the bloodstream, and they bye-bye heart or hello emergency room.
There are a whole lot of Polish people who do the Kwasniewski shuffle ie more saturated fat than you could imagine in your worst nightmare, and they wind up cleaning out their livers, kidneys and intestines, curing a lot of diseases. There are many roads to the top.
I seriously doubt that the ONLY cause for cells being unable to exchange sodium is excess fats. There are probably hundreds of different causes.
You can search the archives for tze tze's intestine cleanse, which was basically just stolen info from the raw paleo people, where people were passing all sorts of parasites and gunk from raw egg yolks.
Also I would say that any condition of low body temperature is related to thyroid function. There might be serious lymphatic blockages that are one of the impediments to thyroid function in that case, but still the thyroid is the body's thermostat. Yes, drinking droppersful of Iodine might not resolve the condition, but that doesn't mean the thyroid is not involved.
Healthy fats have different meanings to different people. If you were a Ray Peatist you would not be eating any avocado or fish oil period, nor any vegetable oils.