There are plenty of video games which have reached a culturally relevant scale. Pokémon is an example; one would be hard-pressed to find a person who has never seen or heard of Pikachu. Even if they don't know what game Pikachu comes from, it has become so iconic that it is recognized worldwide. Super Mario, Sonic, and many others have also become cultural icons over the past few decades.
The majority of society may not see video games as fine art, but that doesn't mean they never will.
There have been plenty of great artists and artworks which were seen as small and insignificant in their time. Vincent Van Gogh was one example of an artist who was heavily criticized and seen as a failure during his lifetime. No one on the streets would have recognized or cared about his art back then, but show anyone today a picture of The Starry Night and they will immediately recognize it. Timelessness doesn't happen overnight; a work of art turns timeless after enough time has passed, and it has become or remains culturally revered. No one today can say whether a certain video game will be considered a work of timeless fine art, as only the passage of time will reveal that.
Video games have the potential to be timeless. They are impactful on society as a whole. They are a new medium of fine art.