How can I find old iCloud Messages on my iPhone 16 Pro?

To view iCloud Messages, you turn on the option on a device that does not have it turned on, and the messages will sync down to it.

click here ➜ Set up iCloud for Messages on all your devices - Apple Support

There is no option to view iCloud messages on the iCloud Website. They are just stored there, but aren't directly accessible from there.

Any message available to you is on your synced device.