I backed up lots of emails that I deemed precious, but I still search through email first, because sometimes it's just easier to search email than to search my backups.

Also, oftentimes I search email not so much for the content, but to find the timestamp associated with a particular event. I have had to search old email metadata a few times when I get an unexpected question related to time (for example, gmail will ask when you created the account as part of its account recovery process).