First, the Kims in Korea is around 20%, which is by far the most common last name but not near the 50% you exaggerate.

Second, the Korean last name system is divided into various clans or bongwans (본관) that have various origins. Some families immigrated from foreign lands, the majority of which is from China. Others derive from families that move to new locations and settle elsewhere from where they originally were from. If you ask a Korean person in Korea, what bongwan they are, they will answer with like "I'm a Gyeongju Lee" or "I'm a Gimhae Kim". The first part of their name corresponds to their family's respective hometowns.

The most common last names like Kim or Lee/Yi/Rhee (romanization has led to lots of variations) or Park usually are indigenous last name that descend from Korean royal families. The lesser common ones like mine are typically from China or Japan and rarely somewhere further away. I think as Korea becomes more and more multicultural we will see more variations on last name come up, though the 3 aforementioned ones will likely always be the most common.

As for the police clearance thing, I think it helps that 1. Korea actually has a good ID system that's assigned from your birth. The first 6 digits is your birthday and the second part is your personal ID number. 2. It also probably helps that women don't change their last names after marriage, which makes the clan system more useful for identifying people. For example, my mom and I have different last names because Korean society is patrilineal. Even if there are two people who have the exact same names, they might be differentiated by either birthday/ID number (which is typically used by institutions).

And one more thing, even if two people are in the same clan, they might not even be aware of it. Some of these clans have millions of members. Even a smaller clan like mine has 160,000 people and I probably know maybe 20 of us at most. The clan thing isn't a big deal, but it's more a legacy of the past kind of thing.

Hope this helps.