I disagree with that!
Red works because he was, for most players, THEM. Symbolically, players of Gen II already became the world's greatest Pokémon Trainer in Red and Blue. They had all eight badges, they defeated the champion, and now they were the best of the best. They were the new standard for the all-time greats.
This is all very dramatic, but again, this is within the context of the universe. While Blue's championship status was short lived, Red's became a legacy, because of the projected talent and nature of the actual player.
Gold and Silver is a parallel journey, but it's also very much the same. You are another ordinary kid with a similar sprite. You are even red. But in your journey, you do twice as much. Twice as many badges, twice as many Pokémon, everything in Gold and Silver is bigger and more. But rather than dethrone Red, the player's previous experience, Gen II maintains Red as the standard. To truly become the best and the greatest, again, you must surpass all of your own accomplishments. Not just quantitatively, but qualitatively and symbolically.
You must surpass the bar you already set. You must defeat Red.
Red is representative. Red is the final challenge that asks the player to prove they have become even stronger and even better than the end game of Gen I. By beating Red, you have finally continued YOUR OWN legacy as a player by redefining what it means to be the best.