The data you can see on your iCloud drive does not include a device backup. These backups are not accessible in iCloud Drive (or Files on your iPhone) and can be "seen" only by an iOS device when it offers them for restoring to a device.
You seem to have set up your new iPhone without being able to restore from an iPhone backup and I suspect, from your description, "I get told that there is no data to restore from my icloud, that you don't have one" that your limited iCloud space has prevented your old iPhone from performing a successful backup.
On an iOS device you can navigate to Settings > [Your Name] > iCloud > iCloud Backup
This is where you turn on iPhone backups. Under the label "Back Up Now" you will see a date of the last successful backup. As your old iPhone is dead you cannot verify this.
Were you syncing any of your services such as Contacts, iCloud Photos and Calendars? If you were these will be part of your account and should be drawn down to your new iPhone when you sign into your account. Likewise, anything in iCloud drive would be visible to your iPhone in Files. Your other data however might not be recoverable.
What does iCloud back up?
- App data
- Apple Watch backups
- Device settings
- Home screen and app organisation
- iMessage, text (SMS) and MMS messages
- Photos and videos on your iPhone, iPad and iPod touch [when iCloud Photos is not activated]
- Purchase history from Apple services, such as your music, films, TV programmes, apps and books
- Ringtones
- Visual Voicemail password (requires the SIM card that was in use during backup)