[CalebD]

Have you ever applied teak oil to a nicely finished piece of teak? I didn't think so. Your main question seems to be: "What is the simplest way to keep the wood looking mostly good, i.e. not completely gray, even if it requires a weekly scrubbing with a cleaning pad or something? I've heard the oxalic acid treatments look great -- just like what I use on my deck at my house -- but that it's bad for teak."
I know that you tried to be specific but do you want the wood to be looking mostly good or not completely gray?
Since you have next to very little exposed teak I have to second or third the use of teak oil for the ruddy brown look and grain enhancing qualities it gives the wood. Without a varnish or other synthetic finish it will not last long though and probably less in warmer climates. As already suggested a simple cleaning with some mild detergent, very minor sanding and application of teak oil will keep it looking pretty uniformly ruddy/brown. You should be able to cleanup any excess oil with mineral spirits or paint thinner and a clean rag.
The oil helps rejuvenate the cellulose fibers of the wood and repel moisture which is indicative of darker spots where mold has set in. Once you have cleaned up your wood (even just a little bit) and oiled it ALL that is needed is periodic (say monthly) wiping the wood with a little teak oil to bring it back to 'like new' appearance, no sanding req'd.
Some people just love the silvery appearance of weathered teak and there is an oil (that does not contain Linseed oil) that allows you to oil it and still have that silvery finish. I just can't seem to find a link for that kind of oil on the web.
Getting your hands a little dirty once a month doesn't sound that bad to me considering it should take you all of 15 minutes to give the teak a little oil.
Good luck with this.