Is there a way to see the time that a photo was taken on the iPhone?
You are not missing anything, the iPhone has no mechanism to see that information natively. However, if you search for EXIF on the App Store, there are a few (some free) apps that will happily do that in the device itself. 

You must note that, due to the way iTunes work, pictures coming from iTunes will not have their metadata associated. Sad but true.