Yeah, the client is a ridiculous resource hog. You'd expect it to be less greedy than the game itself, but no... it's the client that crashes on me all the time, not the game. I can keep the game at medium settings, but I absolutely have to keep the client in low spec mode or I crash immediately upon entering champ select. That's either a forced dodge or getting locked into the wrong runes since the client usually won't finish reloading until the last 5 seconds of champ select (often not even until the match starts loading).
Why does the client have to be so resource hungry? It's not like it needs realtime 3D rendering like the game does. Any animations can be pre-rendered clips. Audio files might be pre-loaded so we get immediate feedback for champ select bans/picks, but things like the animated ban display are unnecessary (we already see both teams' bans at the top corners, so it's not like the animation is suddenly revealing them).
Supposedly there's a new optimization approach they're using for the client which is already on the PBE. Hopefully that translates to a slimmer, stabler, faster client when it goes live.