You misunderstand. There is an international transit zone at Milan. That was not your problem.
Your problem was that you had a flight from Milan to Rome. That is a domestic flight, and you must enter the Schengen area to board it. You do not leave the Schengen area until after you arrive at Rome, where you need to go through exit controls before you can board your flight from there to Seoul.
(The same is true for any intra-Schengen flight whether it is domestic or not. For example, an itinerary from London to Frankfurt to Vienna to Seoul would be essentially the same because of the Frankfurt to Vienna flight.)
There is no way, regardless of airport facilities, to transfer from an international flight to an intra-Schengen flight without going through the passport checkpoint.
You checked whether you needed an airport transit visa (ATV), and you do not need one. But an ATV is not sufficient for the itinerary you booked. Suppose you were from Afghanistan and did have an ATV: the same thing would have happened, because an ATV does not authorize you to go through passport control. To follow this itinerary, you need either a regular Schengen visa or a passport that allows you to enter the Schengen area without a visa.
As it appears to me, there was (practically) no way for me to find out that I would require a Schengen visitor visa (when I didn't even need the transit visa) just to change flights in Milan
Your research, very unfortunately, led you to an incorrect conclusion. Part of the reason for that is the source, which is unofficial. Unfortunately, I am also unable to find (quickly) an official source that is much clearer about the airport transit visa. If you go to the EU site, for example, you have to know that "transit through the international transit areas of airports of the Schengen States" is not good enough for your itinerary. There probably is an official site somewhere that says explicitly that such a visa does not suffice when there is an intra-Schengen flight, but it's not immediately obvious. I also looked on the site of the Italian embassy in London, and that was not particularly helpful, either.
Is there any chance to get my money back or make the agency arrange a new flight to Seoul at a later date at their expense? Or is it the airline's responsibility since they haven't turned me away at the check-in?
Unfortunately, airlines are very good at disclaiming this responsibility. And governments are also very good at disclaiming responsibility for failings in their public information materials. I doubt you will be able to recover anything, as unfair as that may be.
Some online visa-checking sites allow you to enter more than one leg of your flight at once. An example is https://www.traveldoc.aero/. Had you used this site, you would have seen that you needed a visa for Italy.
As pointed out in the comments, even traveldoc suggests that an airport transit visa would be sufficient for the itinerary, which is incorrect. Another commenter suggests turning to a question on this site: Do I need a visa to transit (or layover) in the Schengen area? Indeed, I have found no official government or airline-industry source that says clearly:
- Are you flying within the Schengen area?
If your next destination is in the Schengen area, you must pass through passport control to get to that flight, so you need a regular Schengen visa unless you qualify for visa-free entry.
