Are old iPhones unable to be use SIM cards overseas, and if so, can it be avoided?

When coming into Incheon airport from KIX, I noticed three places selling SIMs.

One appeared to be from a major Korean brand best known abroad for electronics (such as LG or Samsung), though it's possible that the SIM booth was merely next to it. They tried out their SIM on my new phone (iPhone 5S IIRC), and apparently it didn't work, but they had a WiFi router available. Suspecting a scam, or at least a more expensive option, I decided to go elsewhere.

Another place was a convenience store with a large sign saying they sold SIMs. As the store was busy, I decided to avoid it.

I found another booth dedicated to SIM cards and the like, and they were able to set me up. They noticed that my connection had a certain quality when I should have been able to get better (eg 3G vs 4g etc.), and they worked out I had configuration still set up on my phone which was for my Japanese SIM card, and they uninstalled it and got a better connection.

Moral of the story: sometimes one retailer are able to supply something that another retailer apparently couldn't, so consider shopping around.

Is it possible they just didn't have a larger sim card anymore that would fit it? And that simply putting it in an adapter would have fixed it? I am quite sure that given the right size, a sim card will just work in any unlocked phone. New sim cards should still adhere to basic specifications though they may have some new features other than simply being smaller. As Jonathan said, most likely this employee just doesn't know what he/she is talking about.

If your iPhone is newer than iPhone 5 (nano-sim), most (probably all) major 3 Korean carriers (SKT, KT, LG U+) should have SIM available. Can't confirm if you're on micro-sim.

Apart from maybe having to cut down a larger sim to micro sim size, any sim should work in the phone BUT you might not be able to use all the network services.

E.g. your phone only has 2g data (probably), so their 4g network which you're paying for will be downgraded for you to 2g performance.

Usually those guys telling you they can't sell you what you want is an attempt to sell you something else, something like a late model phone...