In the winery, everything is cleaned with caustic. You can purchase it in non-industrial quantities online or at a local hardware store. Caustic is very basic and so removes tartrates (acidic) very well - if you want it to clean quicker, just increase the concentration (1/8 cup or so should be fine). Rinse with something acidic (citric acid solution) to neutralize the caustic and then final rinse with distilled water if you don’t want water spots.
Warning: Caustic will be ‘activated’ by water. This includes water from the sink or the moisture in your skin. Do not get caustic on your skin, it doesn’t feel very nice. Use gloves and prepare the citric acid rinse solution before using the caustic. If you do get caustic on your body, rinse with the citric acid solution to neutralize the caustic (think fight club).
I struggled with a decanter for years. Glass would get stained with the red wine and it was way too much work to clean. I came up with a solution that might work for you. Take a clear, white wine bottle. Take off the label. Steam clean the bottle. Then get yourself a pewter wine funnel–I have one that I must have bought in England 25 years ago and it was unused for all those years. It has a half inch screen that fits into the funnel and catches sediment (removable for cleaning). I pour the wine from bottle one, through the funnel, into the bottle two (the white wine bottle that has been cleaned). If you see any sediment then look into the empty bottle one and see if any remains–if so wash it out. The funnel has a side exit that helps to aerate the wine as it goes through the funnel, down the inside of bottle two. Reverse the process. Pour the wine from bottle two back, through the funnel, into bottle one.
This process takes about 30 seconds. You have a bottle of wine, in its original bottle, that has been aerated, with the sediment removed. The bottle is now acting as the decanter. Sometimes it is perfect to drink now, sometimes it improves over the next day. But this decanter is only used once.
I steam clean (boiling water through the funnel) into the white wine bottle every few weeks, with any hint of odor.
Simply rinse out your decanter with warm water to remove any existing wine. Soak the decanter in warm water. This will loosen up and debris and soften any stains.
I had a number of glasses and decanters that, over the years, had built up detritus, despite using a range of solutions to try and eliminate staining, including vinegar, ball-bearing swirl, even CLR and Draino (obviously with serious rinsing after). Over time, though, deposits continued to build or could not be eliminated. When re-doing our bar area recently, though, I put in a Thermador “Glass Care Center” dishwasher, and gave up on hand cleaning. This thing has been amazing, and has removed the stains from all of our wine glasses and decanters. I highly recommend it.