What does that even mean? If a business exists, it's clearly relevant to some people. If it's paying its bills, not cheating on its taxes, and is there year after year, it's certainly relevant to its community. It serves a purpose to its customers, enough of a purpose to where they give them money. To question its relevancy is to question capitalism itself. This is true regardless of its quality or whether the owner knows more than you about your micro interests.
There are many reasons why you, personally, might not go into one. Perhaps you don't play games. Perhaps you don't like people or wouldn't belong to a club that would have you as a member. Perhaps the owner is an ass or the clerk looked at you sideways. I've got one Yelp review where it said the employees were amazing but they were managed poorly by me. As if my job is not to obsess over their amazingness.
Perhaps you believe there's an inherent efficiency in giving your money to the biggest company you can find, or getting the cheapest price possible. About 10% of products are purchased online, a ratio that does not grow. E-commerce is a niche. Or, maybe your tastes have grown beyond the offerings of a sedate store into the hyper realm of the long tale and the obscurity of vanity press. Nothing is wrong with any of this. The key though, is whether your FLGS is relevant to you, despite the fact that the Internet can make it seem like your universe is the universe.
And is the FLGS dying? Are they shrinking? Are there fewer of them? How many are there? What exactly is a game store? Should we count big box stores that now sell our games? Should we count toy stores? What would dying look like? While we're at it, how big is the game trade? What's the gross sales of all games combined? What's the dividing line between game and toy? Nobody can agree on the answers to any of these questions.
Game store owners are so unique, so divided by their personal ways of doing things, that lumping them in the same group might not even be helpful. They seem resistant to it anyway, which might actually be a strength, like a terrorist cell where nobody knows whose in charge. But one thing is for sure, relevancy is a personal question.