Some games would continue after that into the 5th century, anyway. Still, it was on the way out before the Empire fell in the West. The origins of gladiatorial combat are disputed, but it's tied to the military and religious rites of the Roman people specifically, and long predated the imperial era. It probably ties back into the human sacrifice themes common to the Indo-European religions. It was a practice alien to most of the peoples in the empire, and was used as a means of asserting imperial authority.
In the 3rd/4th century, we have the empire under strain economically which gives real-world pressure to curtail the games. Demographics are shifting and there are a lot of culturally non-Roman citizens in the empire, including emperors. At the same time, Christianity was on the rise. It erased the ancient religious justifications, and created new religious objections. Once Christianity becomes cemented as the state religion it was pretty much done -- Christian martyrs being thrown to the lions in the Colosseum was an old meme even then.