Interesting range of methods. You want to remove oil (which is hydrophobic) from wood (which is cellulose-based, and hydrophilic). Should be easy - the hard problems are separating substances which are alike - e.g. ethanol-water.
Water-based methods are based upon breaking up the oil into little droplets surrounded by hydrophilic molecules (soap method), or chemically reacting the the oil into something hydrophilic (base method). I would have thought water-based methods would swell the wood but you've heard of success with two methods from folks that have used them: Soap (Murphy's oil soap) which breaks up the oil and allows water to wash it away, and strong base (lye in oven cleaner) which basically turns the oil into soap, allowing it to be washed away. I think TSP has both a bit of both the soap and of the reaction stuff going on.
But when you soak the wood in water, doesn't that swell and warp the wood? In a gunstock, which is thick, maybe the size of the wood resists the warp. But in a clock pendulum? I would worry about warpage.
The solvent-based methods should work without warpage. Paint thinner/mineral spirits will probably work as well as anything if there is no gum or oxidized oil.
You are trying to get the oil to dissolve and diffuse out of the wood. To do that, you want low oil concentration in the bulk solvent outside the wood (stuff diffuses from high concentration to low, so if you keep your bulk solvent oil concentration low, the high oil concentration in the wood causes faster diffusion out of the wood). The whiting (or K2R, or Fuller's earth) work because the oil binds to the whiting (or get's trapped in the molecular "cages" of the Fuller's or K2R's diatomaceous earth). This lowers the oil concentration outside the wood. You can achieve a similar effect with multiple changes of solvent. I would:
1) wipe as much of the stuff off as you can physically, with a rag moistened with mineral spirits. If you have problems with some stuff sticking, try MEK or acetone (they are chemically/physically very similar - if you use one, you probably don't need the other).
2) soak the item in mineral spirits (or mineral spirits with a little MEK). You might be able to find one of those foil turkey pans at the grocery store, and if you cover with aluminum foil, solvent loss should be minimal.
3) remove the item, wipe it off, and see if it is oil-free enough for your taste. Repeat step 2 and this step until you've got the oil out. Instead of switching out solvents, you could use Fuller's earth, but it might take a lot of the stuff to capture the oil. Even if you use Fuller's, I'd do a final "rinse step" soak with clean solvent.
4) Dry the item with a rag, and let the solvent evaporate out of the thing overnight. Mineral spirits, acetone, and MEK all will evaporate readily, overnight if you have the item in a warm, well ventilated space. Might want to wrap blotting paper around it and clamp it between to straight pieces of wood to ensure straightness.
You have heard from others that this will get the oil out, I think without subjecting a long, thin wooden item to the possibility of warping due to water contact.
By the way, you should 1) wear chemically-resistant gloves, 2) work in a well-ventilated area (outside, away from other people and animals is best), 3) wear a respirator, and 4) recover your waste solvent, LABEL it well (e.g. mineral spirits 90%, acetone 10%, with some leached oil) and dispose of it at your county's toxic waste disposal site. The labeling part is important - it costs a heck of a lot more to dispose of mystery solvent that it does a known, common chemical (or a mix thereof).
I am guessing you should finish by using an oil-based finish, but I would ask for a critique from the experts on this site. The thinking is that any residual oil in the wood will screw up oil-based stuff less than it would other finishes.
Good luck,
Jim