Warning: contains spoilers for O.
Giving menu options in a game feels like an admission of failure. The creator couldn't find the correct settings themselves and had to leave it to the players to fix it. You couldn't decide on the best controls, so you left some possibilities to choose between. You couldn't make good enough music, so you left an option to disable it. Some things are unavoidable though. Accessibility options are a very good thing; enabling subtitles or whatever (but it's more elegant to cater for colourblindness by distinguishing things through shape than by having a separate mode). But most of the essentials (volume control, graphics settings) are handled by the hardware or operating system; not my problem. (It's nice to support mods, but that's something different; allowing your game to be used as a frame to build other games on top of.)
You often end up needing to make some concessions for people learning to play - easier difficulty levels to learn on, harder ones to satisfy Terry's craving for rapid defeat. I love the elegance of Ziggurat's single mode, but I know the gradual start puts many players off. In Glitch Tank, I ended up adding the 6hp mode after having played it a lot; it's better for advanced players, but way too slow for beginners. In Helix, I've ended up having to add separate difficulty modes.
In a multiplayer game, I feel it's often an error to include an option to play against AI. The way an AI opponent works is just so unlike to how a human plays that it makes for a very different experience - an AI has an advantage over humans from its essentially instant reaction speed, but a disadvantage from its lack of conscious reasoning. Plus the experience of learning alongside an opponent is something I value and (so far) AI opponents can't offer that. I suspect Glitch Tank suffered from this: while I've enjoyed playing against the AI, it's vastly inferior to playing against another human. It's pretty likely that the first time someone plays will be against the AI, because they've just downloaded it and not found someone to play with yet, so they'll get a suboptimal initial experience. And maybe they'll judge it by that, find it unworthy, and never end up playing two-player. In future, if a game is intended to be multiplayer, I'll try to leave it at that - it won't be accessible to people who "don't have friends" to play with, but I see that as a smaller loss than people who would otherwise play it with friends not doing so.
Again with Glitch Tank, I'm not sure whether I was correct to include the turn-based mode. I've said before that I'm glad I included it because some people vastly prefer it, but now I'm not sure. Has it given anyone a worse experience of the game? It seems like a lot of players find the real-time mode alien at first, then switch to turn-based and are far more comfortable with it. It's much more common to see card mechanics and grid-based movement in a turn-based game, so this is a more familiar experience. And then if they never switch back to real-time they're missing out; it's a much more interesting game, partially because of it being a less familiar approach. It was an intentional choice not to save Glitch Tank's options, resetting to two-player real-time each time it's run, giving a constant subtle reminder of how the game is best played.
O has just one mode, and for some that counts against it. But that one mode is bloody good, and omitting options adds to the overall pure minimalist aesthetic. I see it as a sport, and sports don't come with a menu (although they can be modded: hence the inclusion of the Secret Options Menu and implicit support for one-handed/one-fingered/left-handed/non-contact play). To be clear: I'm not complaining about that review; I'm happy they wrote about the game, he's just writing for a particular audience which I'm not part of.
The Secret Options Menu lets you change physical properties of the balls (friction, bounciness, speed), scoring (target, chain bonus), how fast balls appear, etc. When I've told people about it at the start, I've felt it's hurt their enjoyment. Rather than exploring it and discovering less-obvious tactics, they quickly set about trying to 'fix' it: like many of my games it takes some figuring out so the initial experience has an element of bafflement. I think it's entertaining to try out variations, to mod it, but it's better to understand the game well first. I spent a lot of time tuning these variables to find good settings; random values you choose are likely to be worse. This feeds into the idea of "the paradox of choice" (via Electron Dance) - giving people options can make them less happy, even though it's something we say we want and make use of when it's there.
Mm, I don't really have a conclusion here. It's good to have options sometimes, for advanced players, for modding, for accessibility. But you should take care with them, try to get things right in the first place rather than leaving them as options, and try to prevent the availability of options diminishing players' experiences of a game.