Firstly to note that if you use Health Check it will still ask if you want to close the browser so that it can be cleaned.
A browser has to be closed to clean it.
What you should have done was turn off the background running of the Browser and it’s extensions.
See this for how to do that for Edge:
As you have told CCleaner not to show you that ‘Close the Browser’ message again it won’t show it to you and will just skip cleaning it.
If you stop the background running as per that linked post above then Custom Clean will stop Skipping the cleaning - because the browser is fully closed so doesn’t need to be skipped anymore
Just for completeness, or in case you don’t want to stop the browser background running:
The way to get that message back in Custom clean is to reset ALL of your settings to their defaults.
There is a button to do that in Options> Advanced
