Resolve "Banner alerting of project move is showing up everywhere"

What does this MR do?

This fixes an issue with project / group moved messages showing up in the wrong pages.

The root cause of this bug is that flash isn't cleared unless it is actually accessed / read.

Since our JSON / XHR endpoints do not read the flash, we do not set the message for these requests.

Screenshots

Does this MR meet the acceptance criteria?

Conformity

• Changelog entry
• Documentation created/updated or follow-up review issue created
• Code review guidelines
• Merge request performance guidelines
• Style guides
• Database guides
• Separation of EE specific content

Availability and Testing

• Review and add/update tests for this feature/bug. Consider all test levels. See the Test Planning Process.
• Tested in all supported browsers

Security

If this MR contains changes to processing or storing of credentials or tokens, authorization and authentication methods and other items described in the security review guidelines:

• Label as security and @ mention @gitlab-com/gl-security/appsec
• The MR includes necessary changes to maintain consistency between UI, API, email, or other methods
• Security reports checked/validated by a reviewer from the AppSec team

#32198 (closed)