It’s been many years since I last used a wringer washer, but I would think that any detergent that you’ve had success with in your local hard water should do just fine in your wringer washer too, just add a little more for each successive load in which you re use the wash water, and of course add additional water to bring up to the proper level if necessary. Also, when you’ve used up your current supply of detergent, maybe give a powdered detergent a try.
When I took the first load out of the wash water, always the whites, the water was still very hot, so sometimes I would use a large wooden spoon to fish out the clothes, to keep from being burnt. After the first load the water had usually cooled enough to just use my hands.
There also used to be products like Liquid Glove, which was a water repelling cream that you can apply to your hands before any water and/or chemicals are used in cleaning/laundry. Do a Google search for these products, it may help keep your hands from drying out, and is much easier to use than rubber gloves. When I was a Cosmetologist, many moons ago I used to use Liquid Glove sometimes when I had to do several perms in a day and it helped save my hands from chemical damage.
No matter what, I hope you’ll learn to enjoy the Zen process of doing laundry in your Maytag Wringer Washer, I know I always did.
Eddie