I'll admit to having both a certain amount of traditional feelings about such matters, mixed with a certain amount of modern influence, but (outrageous as it may sound) if I had an old ash pack basket to clean before treatment, I'd hit it with a pressure washer. Obviously, you don't want the pressure set so high that it digs out wood, but it can be done and I suspect it will do a better job than trying to maneuver a piece of cloth into all those little spaces. Rubbing with a cloth does about as much to rub dirt into a wet, raw wood surface as it does to lift it off of that surface. I have an uncle who used to be a big-time antique restorer/refinisher and when stripping old stuff he often slathered the piece with stripper and then blasted it with the pressure washer - and these were pieces of furniture that once restored would often bring in several thousand bucks 20 years ago. I never watched him do it, but he claimed that a fairly fast washing would get the gunk and remaining finish off without really soaking the wood.
When our balloon baskets got dirty, we used to take them to the coin-op car wash and blast them with the wand. Same basic deal, using a low-power pressure washer. Worked like a charm. Varnished baskets were then ready to roll, oiled baskets were ready for a new coat of oil, if needed, and unfinished baskets got a cleaning and quickie re-moisturizing treatment.
As for killing the wood, I assume they're using green wood, which probably resists moisture damage somewhat longer than kiln dried wood will, but green wood doesn't stay green all that long and even when green, it isn't totally immune to damage, rot. etc. from moisture. I just can't see any way that the proper prescription for the preservation of white ash is to do something that has been shown without a doubt to create serious (and seriously ugly) problems and eventually destroy it. Aside from this discussion, if someone walked up to me on the street and asked what the best way to ruin a hunk of white ash lumber would be, I would, without any hesitation, say "don't put any sort of finish on it and get it wet frequently".
Deks Olje #1 matte-finish oil is wonderful stuff. I wouldn't put linseed oil on a boat if it was the last finish left on earth (sometimes dries sticky, often turns black with age, actually feeds some forms of growth if the Forest Products Lab knows what they're talking about, etc.) but I'm a big Deks #1 fan and have been using it on various boats and things since the 1970s. I'm not all that fond of Deks #2, the optional gloss topcoat, as I think traditional varnish goes on smoother, so I just use the #1 matte oil finish. I should probably try #2 rolled and tipped some time and see how it does, as it's been a long time since I last tried it. For a Deks#1 matte finish you simply open the can, slop a coat on, let it sit and absorb for a few minutes and then do it again. You continue this wet-on-wet treatment until is just sits on the surface and won't absorb. This may take 5-6 coats and maybe an hour. Then wipe off the excess and let it dry overnight, at which point, it's usually dry to the touch. To build up a similar amount of water protection with most other oil finishes (including Watco and the other marine teak oils) you would be there for a month or more, oiling, waiting days for it to dry, smoothing, re-oiling, waiting more days for drying, re-oiling, etc.
Throw a new coat or two of Deks #1 on once or twice per season, wipe the excess off and you're generally done. Like any oil finish, maintenance is the key. Let it go too long and dirt and water will find a way into the wood's surface. You always want to re-oil before it gets to that point.
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