How do you entirely backup Google Chrome?
While I can't speak to the completeness of the Google Chrome account sync, All of the settings for your Google Chrome account are kept in your user profile.
So there are a couple of ways to proceed

Keep a current backup of at least your user profile using the backup utility of your choice.
Time Machine works well for #1 above as it backs up your user profile by default (in addition to everything else.)
You can use the Migration Assistant to migrate your entire user profile (Apps, too) from your old Mac to any new Mac as long as you have a backup or access to the HD or SSD in your old Mac. This includes the still working old Mac attached to the new one with an ethernet cable.

I don't use Chrome much but if I recall correctly there are preferences and folders that contain all of Chrome's configuration files, cache, bookmarks, add-ons and whatnot located in your ~/Library/ folder. That is the Library folder inside your user profile that is usually hidden from view in Finder.