Speaking as pure theorist here, because I've never compared the saltiness of cooked chicken.
I think that there are two things you will have to think about. The first one is osmosis (water that gets soaked into the meat through cellular membranes), the second one is transport through porous media - like a sponge, the open ends of muscle fibres of your chicken soak up the liquid you put them in. What you want to do is to prevent osmosis into the chicken (because only pure water will go in through osmosis, concentrating the salt outside) and maximize absorption. I will assume that 0.9% saline is isotonic for a chicken, and I think this is a safe assumption, because veterinary saline isn't marked as species-specific. Seeing that your soup liquid is 0.27% saline, you can expect some osmosis to happen. *
Let's tackle the more interesting part first. You want more absorption to happen. The absorption is described by the Washburn formula,
and you want to maximize your L.
L improves with time. Luckily, a soup is meant to be cooked for a long time anyway. But when you wonder whether to give it some more time on the stove or not, more time is probably better. Keep the temperature lowish for less evaporation, and a lid on which will drip back evaporated water.
L improves with lesser viscosity. No way to influence it in your basic recipe, but in a real soup, avoid thickening. So don't use starchy ingredients, or at least, wash them before adding.
Surface tension should be kept high. Again, this is ingredient-specific. No way to influence it in the basic recipe, but it could turn out that some vegetables are reducing your surface tension a lot - I can't think of a soup vegetable or a herb or spice which is famous for a high saponine content, but it could happen. Also, you should give your pots a finishing wash with clean water if possible - detergent and dishwasher finishing liquid reduce surface tension a lot.
Pore size. My intuition says that this should have the biggest effect in the soup case, as you probably can't influence viscosity and surface tension too much. Obviously, the bigger your holes, the more water comes through. A good way to do that is to thoroughly denature your proteins. The first and most common way for that is prolonged cooking time. Second, brining in concentrated saline (6%) and/or acid before you cook will attack the meat surface, again denaturing proteins. However, I don't know how wise it is to use this option, because some of the brine will get absorbed into the chicken, which will leave less space for broth and will increase salt content as a whole. (This assumes that you salt the broth; Brian's idea of not adding salt to the cooking water after brining has merit). Third, you can use meat from the freezer (this is a very likely reason for the mixed results you saw until now). Freezing produces ice crystals, which rupture cell walls. When you use the thawed meat in a soup, there are more holes for water to flow in.
The Washburn formula is for a single capillary. But the more capillaries you have, the more absorption you get. So, what you want to do is to cut the chicken meat perpendicularly to the muscle fibres. And cut it into many small pieces instead of a few big ones.
Now we took care of the absorption, let's look into the osmosis. You can't change the salt content of the chicken's cytoplasm. But for osmosis, you have to separate the two solutions by a semipermeable membrane (the cell wall). Poke a hole in the wall, and the osmotic gradient vanishes when the liquids mix. So everything from the third point on absorption helps you reduce osmosis too.
As for your suggestions: 1. I see no reason why slow or quick heating will change absorption. I guess the products of a Maillard reaction could clog some pores, if you sear before cooking, but it won't happen during boiling. As I said, cook slow because of evaporation. 2. The chicken shouldn't absorb salt, but salty water, see the footnote. Although, if there is a hole in my theory, this is a likely place for it. 3. I doubt it, but if you wash it, you could free eventual clogged openings. 4. Certainly, cook longer.
To summarize: Cook meat which has been frozen, cut in small pieces perpendicularly to the grain, and stew it for a long time.
* I don't see a mechanism for the chicken absorbing a higher percentage of salt than what you have in the broth. I assume that in your "normal" cases, you are left with salt content near the initial 0.27%, and in bad cases, this gets concentrated.
Update As requested in the comments, I am providing an explanation about "cutting across the fibers". Short story, meat is made up of muscles, and a muscle is made up of fibers, or bundles of bundles of cells. You can easily see them in raw meat. You want to slice across them, so their ends are open, as opposed to along.
The good, long explanation with pictures can be found on Serious Eats