Curious - what is your go-to board game?
I've learned the fun of social deception games with friends fades rather fast, as in the replayability feels low. At least in my case, that is.
I don't have a regular group, and so I play with mostly casual players. The intricacies of role-based play is somewhat lost.
IME, at best, it's fun the first time or two, but at worst, it's frustrating to be deceived and can become emotionally taxing.
boardgamearena.com is a great way to try out lots of games.
Incidentally, boardgamearena supports both single-tile and three-tile hands,
Of the classic "gateway" games, I personally would recommend Ticket to Ride or Carcassonne. The former is about collecting cards of the right colour in order to build rail lines across America (or Europe or various other countries), while hoping nobody else tries to get in your way. Carcassonne is a tile laying game where you're trying to build and claim features in a pastoral medieval landscape, which feels like it should be pleasant but can be brutal!
If you want to see how good board games can look, I personally really like Wingspan. It's a game of collecting birds, and includes a huge stack of cards, each of which is a different, beautifully illustrated bird with unique features that relate to the bird's real life traits. It's a bit more complicated than some of the other games here, but also very satisfying when you can figure out how to fit all your bird powers together and create something very powerful.
The other thing I'd recommend is not so much a game as a mechanic, and that's deck building games. The idea is that you start each game with a small deck of cards and play them. Playing the cards generally allows you to buy more cards which you then eventually shuffle into your deck, making your deck stronger over time. I really like these sorts of games because you have such a satisfying feeling of building up cards that work well together and then playing them in a single giant combo.
Dominion is the oldest, and probably the best example of deckbuilding, it's very simple conceptually, but it's very deep, and you can also play it for free online if you want to try it out. There's lots of other variations on this theme though, and I've heard good things about Clank! where you're simultaneously trying to build your deck and sneak through a dragon's lair. If you are a fan of Harry Potter, there's also a Harry Potter deckbuilding game where the players are working together rather than competing, which is really well tutorialised.
Speaking of cooperative games, this is another genre that I didn't really think about before getting into board games. There are a lot of games here depending on what you're into, but Pandemic is a good starting point: you're working together to cure four diseases that are infecting the globe, you need to simultaneously be clearing up infected regions, but also researching cures so you can win the game. It's fun, thematic, and very tense. Alternatively, if you like puzzles, card games, and tense silence, The Crew is a very simple trick-taking card game where you're working together with the other players, but you're not allowed to talk to each other, so the only way you can communicate is by the cards you put down. If you're used to card games, it'll click very quickly, but even if you aren't, the concept is fairly intuitive.
There are a whole bunch of other games that I want to mention but I've already written a long enough comment! I think the best way to get into board games is to try a few games out and see what works for you and what doesn't. There are people out there who exclusively play intense simulations of 1800s rail stock exchanges, and people who just want to place pretty tiles in the right places; there is really something for everyone! If you want more places to try out games, you can often play virtually with tools like Tabletop Simulator or Boardgame Arena, or sometimes there's a local library or board game group or game shop where you can try before you buy.
All those you always have something to do, which is my main gripe with Catan, where some games can be A LOT of fun, and some others, you would rather be watching paint dry.
I think after some reflection I know what you mean about the variance of "stuff to do" from game to game in Catan. We don't play much anymore, as I prefer asymmetry, but I have such fond memories of it, I was just curious.
I find the variance here way too high. Coupled with being a long game, this essentially bums out a few of the players who end up missing out in resources in early or mid-game and then just stagnating and getting bored. When I'm forced to play the game, I've played by taking the average of a set of dice rolls (essentially shaping your result into a distribution that approaches normal as you have more rolls) but honestly I just don't bother playing Catan when given the choice.
I certainly credit Catan for being my cohort's gateway into the world of board games, a lifelong hobby, and for that Mr. Teuber has my eternal gratitude.