To my knowledge the sync setting only appears in Exchange. I just checked my Comcast account and that setting does not appear there and it is an IMAP account. The setting is also not present in iCloud Mail.
This issue is generally associated with not deleting mail in your inbox and it is stuffed with mail. When you add the account to a new device, it will sync your inbox, and can take a while. The same thing can happen to POP accounts if you do not have it set to delete mail from the server after it has been downloaded to your device. Although there are very few POP accounts left.
The easiest thing to do is to log into your mail account in their web interface and delete mail from your inbox there. I'm bad about it, but I usually take the time 2 or 3 times a year to clean my inbox. I clear my work Exchange account annually, because I set it to only sync a month, I don't see as much in my inbox.