I find it puzzling too that Google would choose the oft-used Escape key to exit the full-screen view. Certainly this is a good choice for novices, but a very bad choice for technicians and savvy users, who undoubtedly use Chrome Remote Desktop much more.
One solution I hear a lot is to launch Chrome Remote Desktop in full-screen mode. The problem with this approach is that, if the display size of the remote PC is larger than the display size of your local PC, the content of the window no longer slides automatically as you move the mouse pointer to the edge of the screen. You must use the scroll bars instead (or you can enable the "Shrink to fit" screen option, which can also be problematic).
I agree that the hovering "Chrome Remote Desktop is now full screen" bar is extremely annoying--especially when the display size of the remote PC is larger than the display size of the local PC. Move the mouse pointer to the top edge of the screen to view the upper portion of the remote desktop and you're stuck fighting the hover bar. Since full screen can be exited easily simply by deselecting it from the menu, the hover bar, though cute, is redundant and unnecessary.