Yep, I've cooked organ meat for my dog before we transitioned him to raw. I've also cooked a lot of it for foster dogs with various ailments, since they're often too immune-compromised for raw.
I've boiled a lot of beef and chicken livers (and mixed ground organs) for anorexic foster dogs at death's door with severe, life-threatening respiratory infections -- it's often the one thing I can get them to eat, when they're shutting down, giving up on the fight.
You can also use the cooking water to reconstitute any base mix you might use (like THK). Skim off the fat when it cools and then store the cooking water in mason jars in the fridge for mixing up your THK -- no reason to waste that nutrition.
When I get the organs ground up from the butcher, I have sometimes made organ-rich meatballs (organ mix plus ground heart, tongue or whatever), with organic oats, molasses, and farmer's market eggs -- then bake. I try to keep some of those in my freezer for sick/emaciated foster dogs.
The smell of organ meat cooking is intense -- do it on a nice afternoon when you can open all the windows and put on the house fan. You will have a dog drooling in the kitchen over the smell -- I've never seen dogs go nuts for anything the way they do for boiled liver.
For $2/lb, you aren't going to get grass-fed. However, even grocery store beef is going to be better than kibble beef. It's a spectrum -- don't let the perfect be the enemy of the good, and do the best you can while you find better sources. Call the local producers (slaughter houses) that sell at farmer's markets and to the restaurant trade -- sometimes they'll cheaply sell the unsavory organs humans don't want.
Like you, I don't eat meat and find it revolting. The one benefit to me is that liver, spleen, and kidney is no more revolting to me than hamburger -- it's all gross, so I'm not any more disgusted by the organs that I am by a package of steak. I just pull out the rubber gloves and get it done. I have a separate large pot for the dog's food, separate red-handled, slotted cooking spoons and cutting knives, red cutting board, etc. It's all very easy to identify as the dogs' "meat preparation stuff."