My dear Mr. Main, I believe you've hit the head of the nail of the term "epic."
This term is a term that is much misused in today's society. The word epic denotes scale, a feeling of the small affecting the massive, like in some of the earliest epics: David and Goliath, the epic of Gilgamesh, the Homeric epics, and some more recent ones - such as Lord of the Rings or Star Wars.
Many video games don't really show this. You don't feel like the underdog. Outnumbered, maybe, but not overpowered. You are THE man, the only one that can do it, and you could defeat any odds. Best example, far past anything else: Master Chief. Just think about that for a minute.
Then there's games like Shadow of the Colossus. Wander does not feel like a typical hero. He feels like a kid who is way out of his league, but determined to accomplish this monumental task to save his loved one.
The other part is the weapons. Think of all the heroes, in any (fiction) genre, in any medium. Most of these heroes will have items attached to them - usually one or two. Whether it is a lightsaber, Anduril, or even a pen or a voice, they have one main tool associated with them. Even video game heroes such as those mentioned in the article, like Link or Master Chief, usually have their primary, identifying weapon, and sometimes armor or other secondary item, even though oftentimes it isn't really the main or even the most common weapon you use in the game.
More games need these. Specifically, more games need these to truly be the defining weapon, and have differing uses for the one weapon instead of relying on huge inventories to provide interesting gameplay. Surprisingly, many RPGs manage not to do this, despite the fact that they are the ones that really should the best. They don't even have one at all, most of the time, instead letting you choose your own - but then they don't capitalize on it. These weapons, instead of becoming your own Excalibur, become "Uber Sword of Awesome", with no personality, no story, and no individuality, identical to thousands or millions of others across the world. The main thing they need, even if you choose it yourself, is permanence, a name, and a distinctive style of some sort.
P.S: My favorite example of a truly defining weapon, both in the minds of the fans and in the canon of the game itself: the Keyblades.