I understand this is not what you had in mind for your project and therefore not so helpful but, just for the sake of exhaustivity, most mahjong panels in arcades don’t use any direction per say, let alone up and down specifically. Obviously, the number of inputs is not reduced at all in the process.
Obviously lightgun shooting games and touch-control games don’t use directions either. Technically keyboard-typing games such as The Typing of the Dead do have all four arrows from a traditional keyboard, but another good example of a specific arcade layout that muddles the qualifications is Pop’N Music. In most of the games, the blue buttons actually help you go up and down in the song selection menu, but they are officially labelled as left and right on the panel.
OK, that’s enough for non-helpful examples. Pop’N Music does allow a good segway into more useful answers. The Bishi Bashi series divides the nine buttons of Pop’N Music around three players. So you only need three buttons per player, which would probably fit the kind of mininal input layout you are looking for.
As mentioned by Nemoide early on, most of the early arcade panels did not have a joystick but rather a couple of buttons to move the player’s sprite. So the number of qualified games becomes astronomical if you accept all Space Invaders and Galaxian clones that came out in the late 70s / early 80s. Here is a panel for Nintendo’s awesomely named Space Firebird (I believe this is from the Sega/Gremlin version of the game, which they had licensed for the US).
There are also versions with sticks, but they use two-way sticks that only go left and right. You will find many similar examples in early arcade cabinets. For instance, I believe an early “platformer” like Circus Charlie doesn’t require up or down. Although there are examples of multi-directional games pretty early on in arcades, I think the colossal success of Pac-Man is what truly really “freed” game design from two-dimensional thinking.
Also, you mention specifically left and right, but keep in mind everyone’s arcade grandfather Pong only goes up and down. So you could easily reproduce the controls with customized button assignement. Same for Pooyan, as already mentioned above. (I am assuming you plan to use an emulator and not simply wire in some JAMMA controls with nothing connected to the “up” and “down” pinouts.)
Someone mentioned racing games. I think one really cool example is AM#2’s Hang-On which required the player to move their entire weight left or right to tilt the bike. It does present the same problem as treefroggy ’s potentiometric paddle above : just using digital left and right buttons will not properly replicate the accuracy which the game expects from you. This is the problem you will get with games such as Arkanoid and Puchi Carat. But if this is not too much of a problem for you, you can obviously add all titles from the upcoming Egret II Mini that make use of the paddle; some have already been mentioned above but you have all Arkanoid sequels and Plump Pop to consider.
Regarding more recent games that delibarately used simplified controls, a game with a very similar control scheme to a paddle control game is Peggle. You would lose some precision but the game is still playable in theory. Donkey Kong has a few interesting examples as well. Paon’s Donkey Kong King of Swing on the GBA only uses the
D-pad +control pad in the menus. The entire game is otherwise played with L and R.
And Nintendo’s own Donkey Kong Jungle Blast also has very limited control outputs that could probably be mapped to your theoritical controller.
If you are into simplistic control schemes, you could go further. There was a craze about “one button games” around the mid-2000s, partly fueled by a community trying to make games more accessible to people with reduced mobility or motor skills. I remember covering a few one button game jams for IC’s front page back then. Here is an example of a page covering such one button games. Outlets such as Gamasutra also had discussions on the topic. Here is an article on the topic.