Does "Automatically back up to this computer" in iTunes stop iCloud backups?

That switch controls what iTunes does. The device has its own preferences. Offhand I don't recall if changing the option in iTunes actually changes the option on the device. I'm all in on Apple Music these days so rarely connect to iTunes. If you're anything like me daily iCloud Backups are more useful than occasional backups to a computer. But I still make both when planning to make major changes.

tt2

I'm pretty sure you can enable automatic backups to your computer, and still retain the option for automatically backing up to iCloud. You can also trigger a manual backup at any point. Try enabling automatic backups on your computer, then once complete check on your iPhone under Settings > {Your Name} > iCloud > iCloud Backup. Having multiple backups is ideal in case you find that one won't restore when you need it. See also Backup methods for iPhone, iPad, and iPod touch - Apple Support. Data that ordinarily lives in the cloud, or would be synced from a computer (such as media), isn't included in device backups, but settings and other data is. Give yourself options for recovery, and don't forget to make backups of the data that isn't already catered for.

tt2

iCloud backups typically happen once a day when your device is locked, on charge, and on Wi-Fi. You don't need to connect to a computer. Local backups to a computer typically happen once a day, if you connect and sync.

tt2