EDIT: I just rewatched Game Maker Toolkit's video on
Lives systems, which is essentially what I was trying to say in my post, but polished. In case you wanted a tl;dr. Original post follows.
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My gut instinct was to say "Yes", limited continues are outdated, which is how I voted. After reading the discussion and putting some thought into it, I think there's a little bit more nuance to them, but not so much to not still be essentially "Yes".
The essence of a continue system is to pick up at a checkpoint without losing all of your game progress. I would consider save systems to be a type of continue system. So the idea of having limited continues reads to me like having limited saves; that feels a bit silly to me.
Lives, on the other hand, we can have a healthier debate about. There are absolutely places in game designs where limited lives make sense, and other places where infinite lives makes sense.
But one way you could think of lives are as
non-permanent checkpoints, and (infinite) continues as
permanent checkpoints. If you think about lives and continues both as different types of checkpoints, then their use cases collapse somewhat more nicely.
Instant-respawn platformers like Celeste or Super Meat Boy which feature infinite lives often don't feature continues, because the challenges that you need to get across are built to be permanent checkpoint to permanent checkpoint. Continues are kind of meaningless, assuming that the game lets you save in any safe spot.
Traditional platformers like the modern 2-D Marios and Donkey Kongs feature limited lives, but you get infinite continues. You have a certain amount of lives to get through a level, overcoming a series of challenges, with some in-level checkpoints for intermediate progress. But you don't get to bank your progress until you can show that you can get through all of a level's challenges on a limited number of mistakes. Getting a game over is a relatively minor setback. If the game is well designed, the earlier challenges should prepare the player to take on the later challenges through theme and escalating difficulty in case they need to start over. I think that most people accept games that require this level of proficiency from its players.
But games with a limited number of lives and limited continues are somewhat antiquated in this day and age. I suppose that if a game has limited continues, then there's an inherent nature for it to be short enough to be completable in a single sitting, so maybe there's an argument to be made that it's acceptable for a game to require full-game proficiency. But if there are difficulty spikes late in the game that the rest of the game does not prepare the player for, then a limited set of continues may not be enough before sending the player back to the start, to complete stages with mechanics for which they have already proven themselves. It can sometimes be far too long to get back to where you are compared to games with more liberal, permanent checkpoints.
Sidescrolling (or vertical) shooters or retro platformers (with a smaller number of stages) could do well with a stage select, if they must adhere to a limited continue policy. But then stage select is essentially a permanent checkpoint, and so is analogous to having infinite continues. For shooters in particular, if a continue starts the player right where they left off, the continue is essentially just extra lives. I'm actually okay for shooters to have limited continues in that respect, so as to require some mastery of the game in order to see the ending.
And on that note, I heartily welcome features that either get around a limited lives or continues system, or encourage mastery of the game under limited lives and continues. On the former side, the aforementioned modern Mario and Donkey Kong 2-D platformers feature super powerups for those who need a little push to see themselves through a particularly tough stage. I believe Crash Bandicoot 4 allows the player the option to have either the traditional limited lives per level, or to let players take as many tries as they need to per level. These features allow for additional leeway in player skill for those that need it, without diminishing a healthy level of challenge for players that don't.
As for the latter side, I have to admit the examples of challenge achievements featuring speedrun or low-damage requirements do fit the bill. 1-credit-clear challenges are a standard for horizontal / vertical shooters. Same for hardcore modes that bake in those challenges. But those are fine as long as they're extras, and the main game has the right checkpointing system via lives, continues, and saves that give the best experience to the majority of players.
