A3 was mostly well-received outside out of the U.S., mostly... especially in Europe. There was only a 5-year gap between A3 and A:R and in the meantime we had plenty of Alien(s) comics, novels and video games coming out.
What handicapped and nearly "killed" the franchise was the decision to place A:R about 150 years later in the future, making it clear and unlikely that humans ran into any Xenomorphs during that long period of time as the scientists' only hope to get their hands on a specimen was to clone Ripley. Then add Jeunet's oddball characters and unique cinematography and the world presented yet not elaborated on, and you have a very limited appeal to continue from. People want Colonial Marines, W&Y; "realistic" down-to-Earth almost outdated tech; biomechanic xenos, realistic civilian characters, outer space; not too far into the future etc. ...A:R skipped all of that. The only three instances of anyone trying to continue from where A:R ended is Alien: Destroying Angels, that godawful Aliens arcade gun game with flying facehuggers and chest Bursters, and Alien vs. Predator vs. Terminator. I'm sure I'm missing some, but the point I'm making is that there's is no real interest in the era presented in A:R.
Blaming A3 for the franchise going downhill is just pure A3 resentment syndrome. A3 still left the possibility to go back to the site of the Derelict to try to find eggs, W&Y is still around and the Colonial Marines are still a thing. The only thing A3 did was ending Ripley's story. If you argue that Ripley IS the Alien movies franchise, then yeah - A3 ruined it. But if you think that Alien is more than Ripley, then no - A3 did not end it or ruin it - it opened up for new movies with new characters.