I don't recall airlines checking passenger visas. Either they don't or I'm not observant enough.

They check your passport, and they know if holders of passport from country X need a visa for country Y.

I've traveled quite a lot, and I'm always asked if I have authorization to enter the country I'm flying to.

I've travelled quite a bit (including Russia -> EU), and I've never been asked that.

Maybe they do a fast peek at your passport when letting you to board in the very end, but they can only check if some visa is physically there, since it takes them around 5 sec.