Before upgrade, BigTemplate used to have a feature where jira system admins can control which Jira projects can have access to BigTemplate's templates.
But after upgrading it to 8.16, it appears that feature is removed and now all Jira projects have access to BigTemplate by default and that each individual Jira project admin should remove the templates from his Jira project. This means, everybody by default will see the templates in every jira issue.
We use Jira: 9.4 and BigTemplate 8.16
Thanks, Aron.
I thought, other users might have the same problem and would gain more visibility. I posted the question to Appfire.
Will update the response here.
There is a global settings option to turn off the enablement of templates for all projects, against each template by going to template configuration of BigTemplate.
I missed this in my investigation and vendor support suggested it.