Picks do change, and fairly dramatically. It's just that they do over a longer timeframe than it's easy to keep in mind. It's true that picks will remain somewhat stagnant within a patch, but the champion pool has a lot of turnover. And it's not just from a specific champion getting buffed or what-have-you. Look at Soraka, who has suddenly risen to permaban status without any changes in months!
Remember earlier this season when Taric was the dominant force in bot lane? Or how Lissandra's fortunes (admittedly mostly on EU for some reason) have ebbed and flowed? Lux's rise? There's only a handful of champs that have never seen LCS play, a list that grew smaller recently when Heimer saw play in top lane.
To be sure, there are champs like Lee Sin and Orianna who have remained perennial contenders in the LCS. They tend to be flexible and very high skill cap. And there's champs who aren't going to see much LCS play unless they're just plain overtuned. Amumu and Malzahar, though very powerful, have sufficiently exploitable weaknesses that they won't be able to compete against the best and most coordinated players.
But still, the champion pool isn't as stagnant as it sometimes seems. It has changed radically, even if not day-to-day.