Phantom update software badge - Seirra to Navajo (sorry if repeated)

The red badge can be for any System Setting. It is not necessarily specific or even related to updating macOS. The only reason for pointing this out is that you may have become fixated on finding a way to convince macOS it's up to date when the reason for its appearance might be something else.

Refer to this excerpt from Change Software Update settings on Mac - Apple Support:

"If a red badge is shown on the System Settings icon in the Dock after you update macOS, you may need to take one or more actions. For example, if you didn’t fully set up iCloud, the badge appears on the icon in the Dock; when you click the icon, the settings are displayed so you can complete setup."

Try this, assuming it is in fact related to a macOS update and not some other setting. If "Automatically check for updates" is already selected, de-select it, and then quit System Settings. Then (this is important) completely disconnect your Mac from the Internet: turn off Wi-Fi, disconnect Ethernet, etc, and restart your Mac. Does the badge disappear?

Then, reconnect to the Internet (obviously), re-select "Automatically check for updates" and determine if it remains gone.

That's all I can think of for now. If Apple Support couldn't figure it out I doubt I could come up with anything more though.