Salesforce support told us that this is an error in the documentation, which they will correct.
The "checkPermission" method inside a managed package will only be able to verify the custom permissions of the package itself.
To get around this, you need to implement the "workaround" solution with 2 queries.
I don't fully agree with this choice/answer, but ...
Per the CustomPermission object documentation's special access rules section, one of these 4 permissions is required after the Spring '20 release.
Special Access Rules
As of Spring '20 and later, only users who have one of these permissions can access this object:
View Setup and Configuration
Manage Session Permission Set Activations