This isn't a particularly lengthy theory, but with how the Pokémon world works, I feel like this is the truth. At the end of the first gen games, Red defeats Blue and becomes Champion. Yet, three years later in Gen 2, Lance in the champion and Red is nowhere to be seen.
Generally speaking, Lance was next in line to the title if Red, who had it previously, were to leave, and this seems to be suggested. Like Mewtwo, who is not in the League, Red wandered off in search of becoming a stronger trainer, abandoning his duties to the League, and hence being free to develop his ability at the sport. Red is, of course, presented in the games as both significantly more formidable than Lance in terms of Pokémon, and also as the final quest or the most difficult trainer, and as such there seems little reason to go along with Lance being any stronger, with the game instead going notably in the other direction.
In any case, though, Red is assumed to be the protagonist of the first game, who defeated Lance. They are hence picking up where they left off in that game, and Lance - generally seen as a one-type trainer anyway - is a stop-gap until the player is deemed worthy to take on the more difficult trainer Red. His task was to become a Pokémon Master, and he is following this in a perhaps eccentric fashion, which the game attempts to convey.
Personally, I think that during those three years Lance defeated Red and Red went off to train so he could win back his title. Then Gold showed up. He disappeared after losing because either:
A) He didn't know Gold beat Lance, so if he lost to someone weaker to Lance, how could he beat Lance as he was currently?
B) He did know Gold beat Lance and thus had to beat Gold to regain his title. After failing to do so, he left so he could become even stronger and defeat the person who beat Lance.
The player was only permitted entrance to the place because they had won the League some time ago, which was not considered sufficient to enter it. In that sense, presumably this was noted, and the game sets things up such that if the player could beat Red, they could probably beat the League, which might be used to train for it and hence is not presented as particularly formidable. Obviously, 'Gold's' battles with Red and so on are not League battles, which would take place at the Indigo Plateau, but considered more difficult and significant. The protagonist of G/S/C was, of course, considered to have beaten Lance by then, as Red had earlier, and with similar Pokémon.
Of course, he disappeared after 'losing' in part because he had to heal his Pokémon and so on, and return to the place.
I love comic books so I come from a world of making your own continuity. I always had the idea that like Red caught 149 Pokemon, became the champion then went to capture Mewtwo. But like Mewtwo kicked the crap out of him. So he went to Mt. Silver to train harder and capture Mewtwo.
Mewtwo isn't in Cerulean Cave at this point, so that would seem difficult. Presumably no plot-line with Red there is to be assumed. Because Mewtwo is post-game content, the game is obviously not going to assume in bad faith that the player lost to Mewtwo (especially given the Master Ball hand-out, etc., probably favouring them at that point), but merely that that encounter did not happen on Red's part. This makes sense in multiple ways, including that the Mewtwo quest is hence presumably judged to be a question of the player's own tendencies, at least optimally. It's worth noting that Red's position does mimic Mewtwo's in the first game, and hence presumably they weren't in any particularly notable feud with them, in the context.
And maybe with Mewtwo's awesome psychic abilities he like shattered Red's mind, that's why he doesn't talk. He's never lost like ever, Mewtwo humbled the crap out of him so he's obsessed.
Can you see why the game might not hold this to be the orthodox plot-line for the player-character?
In any case, seems little reason to hold to such a 'continuity,' except for virulent self-hatred on the part of the player.
Anyway, though, other characters do talk like Red, such as the red-haired 'rival,' as well as the 'Psychics,' so perhaps it would seem more reasonable to hold that he had turned into a Mewtwo. He is never presented as mute, but rather as just not talking, which suits their focus on Pokémon battles and is how humans would generally be when they are not talking. Mewtwo can inflict amnesia, as he does on Ash Ketchum (humorously and perhaps slightly maliciously), but presumably Red is literate. It was among other things a reference to the concept of a silent protagonist, which was still in operation, and as such they are unlikely to present this concept in such a degrading manner, nor the player to want to take it as such, unless they really dislike the games. A person can fall down a cliff, or out of the way of a punch, or because they are Satan. Red is, of course, not Satan, although he does have a Pikachu.
In any case, the game would seem more likely to follow the logic, "I never called upon you and received a courteous reception, and then insulted you."
As the Elite 4 are merely a sequence of bosses, this seems mostly decisive in this question.
