Why is Time Machine not saving Google Chrome History file?
Why Chrome history is not backed up

You are correct that the ~/Library/Application Support/Google/Chrome/Default location you reference is where Google Chrome's history (among other things) is stored.

However, (and I don't recall exactly where I originally came across this), that App developers can determine which files (if any) they want skipped by a Time Machine backup and that Google doesn't allow Chrome's History to be backed up by Time Machine after changes made in 2009.

Although I only use Google Chrome occasionally, I personally get around this limitation by manually backing up the five files not backed up via Time Machine. The five files in question are:


Favicons
Favicons-journal
History
History-journal
previews_opt_out.db-journal


Restoring your missing Chrome history

Now, in terms of restoring your currently missing history, you may have some luck if you have a Google Account. If so, you could check whether you had Chrome Sync enabled. If you did, Chrome Sync syncs your history (as well as bookmarks, passwords, etc) to your Google Account so you can use them on other devices.