I agree with you. Being able to work on a bike is very important since it saves you a lot of time, gives you a chance to realize how everything really works and also will give you an appreciation for some parts on the bike, and also make you understand the "engineering" that goes into it.
Yes, it even saves money. Oh, and not only just on repairs, but it could keep you away from buying expensive parts, so that they might "perform" better.
I am in my late teens/young adult and only started cycling three years ago. From then on, I have learned how to do everything on my bike. Don't do the stuff that requires expensive tools, but everything else is now easy.
Yes, before I depended on my bike shop for everything, but learning how to do things myself, gave me an appreciation for the "cheaper" parts. A bike shop doesn't have enough time to execute things to perfection. While I do. I pay the money for the part, and I make it work. I only have to work on one bike, the bike shop has to work on many. See the point?
But this puts the bike shop into a bad spot. If everyone fixed their own bikes, then bike shops wouldn't be making any money. But there are many people out there that will never even try, or will get bored of trying. Many... So bike shops will always have those people, and the newbies that are still learning. And the people that really have no time. Kind of isn't their fault.
Wow, I kind of said too much. Patience, common sense, logic, and a methodical pace can make it easy to learn how to fix stuff on a bike.