because USB reads and writes a lot slower your patching would be a lot slower and so would your loading speeds (getting into game)
Also, you would run into issues of the computers the game is running on needing certain files and dependencies, such as direct x (the league installer has you install such things during the league install process). Without such things, the game cannot run.
Most library/school computers will not have these installed by default because these computers don't really need to run games.
in other words, the only truly portable version of league would be if you installed windows onto a flash drive, and then league into that. This would only work if the computers had low security, and you would be able to restart, and then choose your usb as the device to boot into.