Cannot Click on Name under settings on iPad

Perhaps, for additional security, you have restricted access to your iCloud settings:

Settings > Screen Time > Contents & Privacy Settings > [Allow Changes to] Accounts >

• When this setting is set to Don't Allow, it is not possible to access or change key Account settings (such as your Apple Account and iCloud) from the protected device.

• When this setting is set to Allow, your Apple Account and associated iCloud settings are available from the device.

This security setting is usually used in conjunction with settings a Screen Time Password - this being independent of your device Passcode. Once a Screen Time Passcode set, changing Screen Time settings isn't possible without knowledge of the Screen Time Passcode.

Subsequent to performing a system software update, settings can sometimes inexplicably change - and differ from the indication(s) within the settings App. As you have demonstrated, toggling the affected setting can restore the expected setting to match the visible control.

Given your description of Apps and services not syncing between devices via iCloud, try a forced-restart - as this will often clear an isolated or post-update software issue:

Force restart iPad (models with Face ID or with Touch ID in the top button)

If your iPad doesn’t have the Home button, do the following:

1. Press and quickly release the volume button nearest to the top button.
2. Press and quickly release the volume button farthest from the top button.
3. Press and hold the top button.
4. When the Apple logo appears, release the top button.

If the problem persists after performing the restart, it may be necessary to reset your iPad settings. While this reset will not cause any loss of data, as a precaution, it is always wise to ensure that you have a current iCloud or iTunes backup before performing the reset:

How to back up your iPhone, iPad, and iPod touch - Apple Support

When you are happy to proceed, you can perform the reset from iPad settings:

Settings > General > Transfer or Reset iPad > Reset > Reset All Settings

If the problem still doesn’t clear, the next step is to reset the iPad to factory settings:

Restore your iPhone, iPad, or iPod to factory settings - Apple Support

You are very welcome.

You may find that simply signing-out from iCloud, restarting your iPad and signing-in again, may clear the sync' issue.