Move bubble icon / content loading into its own controller class

I think this simplifies the BubbleBarController class and makes it
a bit easier to deal with the optional overflow (done in other CLs).

Flag: com.android.wm.shell.enable_bubble_bar
Test: manual - add some bubble to bubble bar, check that the overflow
               is there & can be opened
Bug: 334175587
Change-Id: Ie17fa0279a981a019d5d50b9f71dc87f49d98dee