I have fixed a couple of washing machines and other appliances over the years. It's satisfying although likely not money-saving unless you don't put a value on your time. But I like working on mechanical things so it's closer to a hobby for me.
Being able to repair my Sears Kenmore 80 Series washing machine from the late 90s has definitely saved me money. I've made exactly two repairs in the last 15 years, each costing me less than $50 dollars in parts, neither taking more than two hours to complete, and both saving me from having to buy a new machine. If that's not money-saving, then I don't know what is!
An example is bike repair. If something breaks, getting it fixed would mean driving it to the shop and back again. Doing it on my own terms and schedule, I can fix it at night, often in a few minutes. If it needs a part, I ride a spare bike until the part arrives.
I decide on a case by case basis. I don't change the oil in my car because I can drop it off on my way to work, and in that case, I am getting paid while the car is being serviced. Also, some kinds of work involve more tools and space than I want to own. And mess.
The home furnace, in the middle of winter while my elderly parents are vising... yeah call the guy.
So you're paying for a new transmission, plus labor and service call fee. This easily could exceed the value of the washer, and certainly would have many people thinking "Eh, it's 15 years old, I'll just replace the whole washer."