Times have changed.
In 1963, you could repair anything. These days, because of SMT and the low cost of manufacturing, things are mostly replaced vs. repaired. Not that they can't be repaired ... it's just that it's not a cost-effective operation. In the case of the motherboard, repairing it might cost more than a new one and the new one will be better and faster than the original.
About the only time something like a motherboard is "repairable" is if you consider your time to be free. To make such a thing a commercial operation would be a financial disaster. You may notice that there aren't nearly the number of radio/TV repair shops around that there were 30 and 40 years ago. Unless a TV is an expensive one, such as a big-screen or digital model, it's usually cheaper to buy a new, improved model than it is to fix the thing.
Troubleshooting skills are learned mostly by doing and experiencing. It's something that's difficult to teach from a book, but at least that would be a start. If nothing else, I'd suggest learning to draw and read block diagrams to help you learn signal flow. That can help you learn to isolate problems to specific areas.
Dean