Thank you for answering my questions, this helps me further assist you with a solution.
You are correct, the Adjustment Program is run through WIN only, and Yes, this is the absolute easiest way to introduce Flush into your system. Same goes for introducing Pigment back into the system after flushing.
Your R3000 uses an internal ink line system, very similar to the 3880's. The ink flows from the cartridge, through an ink line, into a Damper (aka Ink Selector Unit) then into the head when ink is called for. Think of these dampers as your filters for the inks, they work in a similar way that an oil filter works in your car, they filter out the large materials so it can't make it's way to the print head. If there has been an accumulation of pigment build up over the year(s) then you start seeing clogged nozzles, because the damper(s) are doing their job so well that it is now restricting flow into the head itself.
Because this is such a common problem with the R3000 & 3880 printers, Epson recommends replacement of the Ink Selector Unit every 12-18 months to keep your printer running at 100%. As of now, your printer is out of that spec by 6 months and if it sits for any amount of time (a week-4 weeks at a time) then you are more apt to see these problems arise early in that time period.
To determine if there is air in your line, which I think may be why you aren't getting any thing printing out of your LK channel, look inside the printer at the ink lines themselves to see if you can visually see any line that might have air in it. You can also try purging that channel with purge images, this will take some time and patients. I would think you would see some improvement after 15-20 us letter sized sheets of the pure LK image. You will be able to determine if the ink is actually leaving the cartridge and traveling through the ink line, damper and to the head by keeping track of the ink level in your cartridge visually. Follow the instructions here for purging an individual line using QTR Print Tool http://www.inkjetmall.com/tech/content.php?166-Flush-individual-channel-using-QTR-Calibration-Mode