You misunderstand. The lack of an international transit zone was not your problem. Even if there had been an international transit zone, you would not have been able to remain in it.

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. (Entering the Schengen area, to be sure, means leaving the international zone.)

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 or any other country whose citizens need an ATV, and you 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:

1. 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.

The cause of your problem is that your route included an internal Schengen flight. This has nothing to do with there being (or not) a transit area in Linate. It is because of your Milan to Rome flight that you needed a visa, not to change planes in Milan.

Airports in the Schengen area are divided in two zones. Schengen, and non-Schengen. A flight from London to Linate will arrive in the non-Schengen zone, flights to Rome will leave from the Schengen zone. To pass between those zones you need to pass an immigration checkpoint, and it is there that you were stopped.

Had your transfer been between two international flights you might not have encountered this issue.

Transit is usually if you're not entering a country, to go onwards to another one. For international flights. Flying between two cities in Italy would be a domestic flight, and you'd need to be allowed into the country (ie have an appropriate visa). Like Almaty to Astana - everyone on the flight is legally allowed to be in the country, because at Astana (or in this case Rome) you're at a domestic terminal and could just walk off the plane into the city (and country).

I would feel that the airline is meant to check this as well, as they can get fined, but I suspect ultimately they're going to say it's the passenger's responsibility.

Awful that you were treated that way though :/ Although "barely spoke any English" is hardly their fault in Italy. :/

I don’t see any chance for a refund. It’s a passenger's responsibility to have all needed visas for a journey and you do need a Schengen visa for a Milan-Rome leg which is a domestic flight, and thus, you would have entered the Schengen zone.

So sorry -- that sounds truly horrible.

It's reasonable to expect a passenger to be aware of and comply with all requirements for entering the country of their final destination.

It does not seem reasonable to expect a passenger to be able to spot that a leg in an automatically generated international flight includes a domestic flight that requires an additional visa. And I really don't think that many passengers would have. If you had purchased the legs of the trip individually, that would be a completely different matter, of course.

I'm certainly no lawyer, but generally worded disclaimers, like the one netflights.com is using as grounds for refusing to reimburse, cannot override laws and regulations related to consumer transactions. Under "good faith" laws, the consumer has the right to expect that rendered goods and services conform to what was purchased.

Booking an automatically generated international flight, I feel like it's reasonable for the customer to expect that the ticket will get them to the destination if they meet the requirements for entering the destination country. Unless very explicitly having been notified otherwise.

So the service you purchased, getting transport from A to B, had a serious issue with it, and it wasn't reasonable for netflights.com to expect you to be aware of the issue and accommodate for it.

Sadly, getting a lawyer involved doesn't seem like it would make much sense financially. But I think it would be worth contacting official agencies in the UK that concern themselves with matters of consumer protection and consumer rights with your story and see if they're interested in looking into it.

Talk to a lawyer, or the Citizen's Advice Bureau. You might be able to sue them for selling you a defective product and refusing to give you a refund.

The Consumer Rights Act (2015) requires that all goods sold in the UK be of satisfactory quality, fit for the consumer's particular purpose, and as described by the seller, and further mandates that if they are not, then the consumer must have a period of at least 30 days within which they can reject the goods and ask for a refund or replacement.

Since the goods they sold you were no fit for your purpose, and when you asked for a refund, they have failed to give you one, they seem to be breaking the law. As such, talking to a lawyer about remedies seems to be a prudent course of action. The Citizen's Advice Bureau provides free legal advice on consumer rights issues, but you may want to pay for a lawyer instead.