For information of those who come to this useful thread:
When we don't see "Save to Files" after "SEND" -> "Share", just swipe the window upward. Then "Save to Files" and other functionalities will appear.
This symptom starts to happen since iOS 14.7, I think, as I didn't see this problem on 2021/7/19, but met this on 2021/8/10 and later. Apple's "Apple security updates" page tells iOS 14.7 was released on 2021/7/19 in my region.