Before the boat traffickers set up, people would obtain false papers and book a flight. Operationally the airline will likely check you have all the necessary paperwork, because they can be held liable for your costs. This is spread among various piece of aviation law The Carriers Act 1830 isn't relevant.
I haven't flown for many, many years, and it is even longer since I needed a visa. I probably wasn't even aware that the airline were checking my visa.