I think the real question you're asking is which skills professional chefs are taught are worthwhile to learn as a home chef?
Professionals have to do all kinds of things that you'd almost never do at home such as cook for 50-100 people every night, maintain huge quantities of stock, soup, ingredients, partially prepared/prepped/cooked food that can be finished to order quickly, etc. Basically be able to serve anything from a rather lengthly menu in 10-20 minutes. This is completely different than the typical home chef that only cooks 5-10 times per week and can plan in advance what you're going to have.
Also a lot of restaurant food is really intended to be extremely flavorful, "special", suitable for a night out, etc. which often means high in calories, salt, sauce, etc. This is great if you're going out to a restaurant but you would get sick of eating like this every meal, talk to people who travel for a living and their number one complaint is the food. Often you just want relatively simple and fresh food and just the quantities you want, rather than a big fancy restaurant presentation. So a lot of restaurant preparations don't translate too well to the home chef.
I think the best skills a home chef can adopt from a professional is quick and efficient preparation, being able to multi-task and time different dishes so they will all be ready at the same time, and learning how to change and adopt recipes based on what you have on hand rather than needing to make a special trip to the store for every meal.
This is why I am a fan of chefs like Jacques Pepin who have been professionals for a long time but can demonstrate how to cook a 2-4 course meal for 2-4 people in about 30 minutes based on ingredients that most people have on hand. This is the best and most practical, not to mention healthy, cooking that home chefs can do IMHO.