I always caught the Red Gyarados, though I don't think I ever really bothered training it. For me, if a Pokémon could be seen on the overworld map prior to the battle (as in the case of Gyarados, Sudowoodo, and legendaries), it was worth my while to save first and battle it as many times as necessary to catch it. In Gyarados's case, I think I caught it the first time, though.
I do believe I dragged it along with me when I went to battle the Elite Four, but that was because the only Pokémon I had was a (fairly weak, unevolved) Totodile when I first played through Gold. I think it mainly served as attack fodder while I revived/healed Totodile.