Question Half broken pin on motherboard's CPU socket ?

Unfortunately most modern electronic is not repairable for a reasonable price and doing it your self tends to also not be economical because of the price of specialized tools.

If you are willing to spend some time doing some very tedious research you might find that you do not actually need that pin. There are diagrams of the pin layouts for cpu chips. A huge number of them are grounds and/or power. Many time you can run without all these. Data pins on the other hand are likely all required....but maybe they only run say the second set of memory slots.