I think that sometimes we, being veteran boardgamers, can be snobbish and condescending without realising it ourselves. The very notion of gateway games is in a way condescending. We tend to think of non-gamers as muggles. I think most veteran gamers tend to like heavier games, and treat simpler or shorter games as mere fillers. When given a chance to play games with non-gamers, we often have the mindset of picking these easier gateway games to play with them. Is that really necessary? I can understand picking something simpler makes it easier for the non-gamers to learn the game. But then they may not find such games interesting, and they may end up thinking that boardgaming is not very interesting at all. It's probably better to consider the type of game the non-gamer will like, rather than emphasising the ease of learning.
When I first played Carcassonne, just before I got into the hobby, I found it very alien and I didn't really enjoy it. At the time I only liked games such as Axis & Allies (1984 Milton Bradley edition) and Samurai Swords. Carcassonne was simply too different. Today, many years after that first encounter, I like Eurogames a lot, and I like Carcassonne a lot too. But I still remember that uneasy first experience.
Recently a colleague who has just started buying boardgames (partly influenced by me) asked me about Railways of the World (new version of Railroad Tycoon, which is quite similar to Age of Steam and Steam). The first thing that came to my mind was whether she was ready for it. When I reflected on my reaction, I wasn't so happy. Why do I automatically assume new gamers or non-gamers won't be able to handle a medium or medium-high complexity game? I wonder whether I seem condescending (at least from their viewpoint) when I discuss boardgames with non-gamers. I hope I don't. Most gamers hope to bring more people into the hobby. If non-gamers perceive us as snobbish and cliquish, then we are creating barriers for others to join us.
In recent years my urge to "convert" friends to gamers has diminished a lot. I have had very little success, unlike when I first got into the hobby in 2003 / 2004, when I was in Taiwan. I rarely actively try to invite non-gamers to game sessions. I still cross-post my blog posts to my Facebook wall, to establish myself as the go-to person to find out more about boardgames among my friends. Nowadays I feel that the right way to bring people to the hobby is just to mention it in passing. Those who have no interest will not ask much further. Those who are interested will ask for more information, and only then you can gradually share more about your hobby. I find that sometimes it's hard to hold back the tirade when you find a potential convert, but I think if you try to "sell" your hobby like a fishmonger at market closing time, you'll often end up scaring away your customers. So, quiet-guru-but-always-willing-to-help is probably the best approach. Think of yourself as a user-friendly walking encyclopedia.
As for picking games for that first session or first few sessions, I think the best way is to go for something the new players can relate to. E.g. Barbarossa for people who enjoys trivia games, Axis & Allies for people who have played and liked Risk before. Theme is also a consideration. You can consider jumping straight to Age of Steam if your friend is a big train fan. Generally, you probably don't want to go for something too complex or too long. No Die Macher please. But I think medium complexity games are fine, e.g. Puerto Rico, Battlestar Galactica, Princes of Florence, Amun-Re, Agricola. The new players may not do well, but at least they will hopefully enjoy the process of learning the game and discovering what the fuss is about.
Please treat the above as opinions and not advice. I have not been exactly successful with converting non-gamers. Maybe I'm just not lucky enough to know many people who are interested in boardgames - not knowing / meeting the right people, as opposed to not applying the right techniques.