General question about visas for connecting flights booked via travel agency

Just ask yourself, where would they get the necessary information from and do you want them to have it? By default they only have your name and billing address, which tells them exactly nothing about visa requirements.

They might get a nationality if they require a passport. This might be enough to have a good guess in the simplest cases, but they might not know about your second passport. They will surely not know that you recently travelled to some place on some exclusion list. They definitely shouldn't know about any past immigration problems or your criminal history and about whatever else there is on the long list of criteria that determine whether you need a visa or not.

Thus the only realistic answer they can give you is "You may or may not need a visa, please check for yourself.", which I am sure they do somewhere in the terms and conditions.

It is always the responsibility of the traveler, that presents themselves before an immigration official, to have the needed documentation to pass through the immigration process.

This is true even in cases where the carrier (such as airlines or ships) can be held liable, sometimes with fines, for the transport back to the original point of departure.

There may be cases, where a traveler may be rejected for personal reasons, which the carrier cannot know or judge beforehand.

In Europe, when immigration controls were common place, there was never a control beforehand.

If a person, traveling on a train or ferry, was rejected - they would be detained and placed on the next train or ferry going back. It would then be up to the carrier to collect any fees for the return ticket.

It is always the person who travels who has the final responsibility. It also says so in many (most/all?) travel agency contracts I have ever seen and it is also a situation we have seen here on Travel Stack Exchange many times.

A good travel agency will point it out to you if you need visa, but most travel agencies will only know the standard for the country they are in. That is the people of that country and often the most common foreign nationalities.
Very good agencies will also check on a visa website but they will only do that as a service, and will (usually) not take responsibility.
And most travel agencies will realize they are not visa specialists and will tell you that you have to check (and often how) if you are not a national of the country you make your booking.

On Travel Stack Exchange we mostly refer to one of the database sites where the airlines also check whether you need visa. (You find links to those in the accepted answer.) Many of the travel agencies in many countries will also use one of those, (often the portal used by the national airline,) but others use sites which are not kept up to date.
So best double check, even when the travel agency has checked.

Even when you have done everything right, visa requirements can change overnight.
At one time my brother, his girlfriend and I wanted to check-in in France for a ferry to the UK to find that the girlfriend needed a visa, started at midnight that day. And announced less than 24 hours before the start of the new rule, (I believe less than 8 hours before the start but anyhow, we missed that information as she was not in her home country.)

Added based on your added details:
Airlines will usually not make scheduled stops in countries or areas where most of their passengers will not have visa or rights to stop over. And the passengers which are on the ongoing plane, either staying in the plane or being 'kept' in a closed of area of the airport, so said passengers can not enter the rest of the airport and thus not the area/country.
In case of an unscheduled stop they will make more certain that all people on the flight are allowed in or kept away from entering the country. (Often done by immigration rather than the airline.)
In any case, if a stop on an airport is announced before you get on the flight, it is up to you to check that it is one where you can go with your passport and visa.
We had a question on TSE a while ago where someone flew from one Non Schengen European country to an other, with a stop and change of plane within the Schengen area, but on an airport where they do not have the option of staying in an 'international' zone. She was rejected at that airport and put on a flight back, with having to pay her own ticket. Connections with change of planes in two Schengen airports are often offered but can only be used by those who do not need a Schengen visa or already have one. Travel agencies often forget that fact.