Photos Won’t Download After Transfer

iCloud Photos is not part of iCloud Backup. If you use iCloud Photos, then having pictures there and also in Backup would be redundant.

You don't say how large your Library is, but it can take quite a while for a Library to download-- days to weeks, depending on size. Not only must the pictures and albums be shipped over the internet, but also thumbnails and previews and the database. And, as the pictures come in, they must be scanned for contextual content, like faces, etc.

Most of this cannot happen while the phone is being used. People like to complain about their batter life, so the iPhone will avoid syncing when not plugged in. And since syncing uses processing power, and people hate slow downs while they play games and watch cat videos, the iPhone will avoid syncing while it's being used. And since syncing requires bandwidth, the iPhone will avoid syncing when the internet connection is weak.

So, if you want faster syncing, then find good wifi, plug the phone in, and leave it alone. Over night is good.

What does the syncing status indicate? Click on that blank pink thing you blacked out, your Apple ID icon, and see:

Mine says "Synced with iCloud Just Now."

Is it correct that you see all the pictures you expect at iCloud.com?

Have you tried a forced restart? 

Force restart iPhone - Apple Support

This is different from a simple Power Off.