Keeping it brief so I don't go far off-topic - definitely that helped them, though the social mobility you describe is not actually that unusual in the ancient world (it's later periods it becomes unusual in), nor in practice did it work well at all times (you needed informal outside-the-system help to get beyond a certain point), but yes that was absolutely the carrot part of their carrot-stick approach to destroying cultures root and branch - abandon your culture wholesale to become a Roman citizen and reap the benefits! But fail to abandon it, and you'll be limited in how you can go. Also the "social mobility" was propped up by massive slavery which enabled it (that's a whole other discussion though).