It doesn't matter how technically impressive the game is --- as long as it's fun that's all that matters, right? Well, yes and no.
At their core, videogames are software. Software cannot exist in a vacuum without hardware for it to run on. If software is the mind, hardware is the brain. Without the mind to give the brain action, the brain is simply a lump of matter. Yet without the brain, the mind cannot exist. The quality of the hardware that the software runs on affects what is possible and to what degree. Videogames are an inherently visual medium, so the clarity and quality of the graphics affect the game experience. Realism can teleport the player into fhe center of a battlefield, while exaggerated art can make a profound visual statement or create a unique, fantastical world.
And it's not just about the graphics. It's about giving developers the environment and tools to create the experience that they want. The more advanced and well designed the hardware, the less restrictions there are on what the developers can bring to life. Super Mario 64 wouldn't be the same game without the Nintendo 64's technical muscle and (at the time) innovative joystick. And while The Last of Us could probably have been developed on the PlayStation 2, it would be severely scaled back and likely not resemble the same game.
The Legend of Zelda: The Wind Waker is another great example of hardware allowing for a stylistic vision to be attained. Though the cel-shaded, cartoony art style was initially criticized, both gamers and critics alike hail the game as one of the most beautiful of it's era, and the art still holds up today. The graphics aren't realistic, but the amount of detail and visual effects included in the game couldn't be done on the Nintendo 64 --- at least not without severely compromising the visual style. The horsepower of the Gamecube allowed Nintendo to bring the style and environment that they wanted to life.
Hardware isn't just graphics. It's the introduction of hard disks in consoles that have allowed developers to improve load times by caching content from the disk and from having games installed on it. These same hard drives allow for games that downloaded from the internet to be stored. Networking allows for players to play online with and against each other. Larger, speedier RAM allows for not only larger textures to be used in games, but in allowing for limited multitasking on modern console operating systems. Multi-threaded and multi-core CPUs allow for more things to go on in the game world simultaneously.
It's also more than just raw compute power. If developers cannot tap into the console's hardware, games may just not run well. Perhaps the most infamous case is the PlayStation 3. Despite impressive exclusive games such as the Uncharted series and Metal Gear Solid 4: Guns of the Patriots, third-party games were often inferior to the Xbox 360 in both resolution and performance. The Elder Scrolls V: Skyrim in particular could end up being relatively unplayable after players invested hours upon hours into it. This wasn't just about the game looking prettier --- this was the game experience being compromised due to complicated hardware.
Perhaps to some people's surprise, Nintendo has been historically very efficient with creating cost-effective hardware that also doesn't limit the developer's vision (the Wii and Wii U being exceptions, despite the Wii reusing the Gamecube's architecture). The late Gunpei Yoki pushed forward a hardware philosophy known as "Lateral Thinking with Withered Technology". By exploiting mature (often times old) and cheap technology, the costs of the console is decreased, and the hardware is already often times well understood. Developers are encouraged to find new ways to exploit and use this known technology.
This was the case with the Nintendo Entertainment System (NES) and Super Nintendo Entertainment System, which utilized CPUs based on the MOS Technology 6502, which debuted in 1975 (for reference the NES and SNES debuted in Japan in 1983 and 1990 respectively). By utilizing this cheaper, more well understood technology, developers didn't have such a steep learning curve in learning and relearning the hardware. This also allowed Nintendo to focus and innovate in other areas, such as the Picture Processing Units (PPUs) which helped create smooth-scrolling 2D graphics, and add further processing hardware as needed through additional chips within the game cartridges.
The advancement of hardware has even led to the growth of certain genres. Though first-person views had existed since the 1970s, 3D graphics in games such as DOOM and Quake helped in expanding the possibilities of this viewpoint. By pushing what PC hardware could do, both games improved on the gameplay mechanics of first-person shooters. DOOM and Quake both ran smoothly and were (and still are) fluent to play. The hellish, claustrophobic level design of DOOM that created such a memorable, engaging atmosphere was thanks in no small part due to the graphical advancements by John Carmack.
Yes, a fun game doesn't need to be technologically and graphically groundbreaking, but it needs to be playable. Players need to be able to understand visual cues and be absorbed into the game world. As long as hardware can be improved, and as long as there are limitations to what a developer can create, the advancement of hardware will continue to be an important part of the videogame industry.
Header image of the Xbox One X internals courtesy of NeoGaf