Are the Schengen "airport transit visa" even checked?

Airline staff can, will and in some cases must check that you have the right visas. They can and will prevent you boarding the plane if you do not have them. As you say the airline is responsible if they allow you to fly to destination for which you do not have a visa.

The likelihood is you will be denied boarding if you do not have the right visa. I know people who have been denied boarding under these circumstances.

Airlines can and do check whether people meet some basic requirements to enter their destination. Of course, they can't guarantee that someone will ultimately be let in (e.g. they don't know how the landing interview will go and have no access to the databases border guards have) but they do have to check that you at least have a passport and visa that look valid.

If your citizenship gives you the right to enter many countries visa-free, that check is completely transparent and you might have not noticed it but for less fortunate people it means they have to present a visa or some additional documentation for virtually every flight they take.

Now, airport transit visas are a special case. If you look at the list of countries whose citizens need an airport transit visa, e.g. in the Schengen area, you will see it mostly comprises troubled countries from where many people try to seek asylum in Europe. Those are also the countries whose applicants have the highest success rate (at least in some destination countries, there are huge, almost absurd, differences between European countries in this respect).

Dealing with a frivolous asylum application costs money but if it's obviously unfounded and the person comes from a safe country, she will usually be detained and removed relatively quickly, at least if she has a passport and is forced to lodge her application at the border. Consequently, people who have absolutely no shot at a refugee status usually don't try that (many many applications nowadays come neither from genuine refugees nor from people stuck at the border, but from people who entered legally and try to delay deportation by any means available - including hopeless asylum applications - after their status changed).

Therefore, what this requirement is designed to achieve is to prevent people with a genuine claim to asylum from reaching the territory of a country. Because once you are there, the right to see your application examined and, if you meet the criteria, to be allowed to stay in the country is an established principle in international law (“non-refoulement“) that European states mostly honour. But there is no obligation to assist the people who are stuck in their region of origin, so we had rather have as many of them stay there instead of getting an opportunity to apply for asylum.