I always look forward to the Lake of Rage and catching the red Gyarados, I think it is one of the coolest plots of the Johto region games. I sure did snag him with wonder in my eyes, and add him to my team when I first played Pokemon Gold. he was a strong and unforgettable part of my team, right there with me when I fought the Elite Four.
The red Gyarados also on my team again in Pokemon soul silver. He was quite fun and colorful to have walking behind me in the overworld. He was at my side when we took on Red.
When I played Crystal on the virtual console I picked Totodile as my starter. In Gold I chose Cyndaquil, and Soul Silver I used Chikorita. Since I had a strong, reliable Feraligatr on my team I didn't really didn't have a way to incorporate Gyarados long-term, but I still wanted to catch him. I consider him a Pokemon well worth transferring over to Sun and Moon. He also still got to see some action on the VC because I liked this Pokemon, and wanted to see him win a few battles. I used him to defeat Team Rocket, my rationale being that he was a Gyarados out for revenge against them. He was able to enjoy a break, chilling at daycare, where he became the respected father of a line of shiny pokemon. In Gen 2, shiny parents have an increased chance of passing it on to the child.
The red Gyarados is a special pokemon. Why not take him home with you? At least go surfing on him. I could only see myself knocking it out and not catching it if I was doing a monotype or a challenge with rules that would prohibit this pokemon.