Hello Shane,
it would be great to be able to remove/ delete or disable (unpublish) the built-in CORE composer templates
Do you mean that you want to disable some of the core post templates in easyblog? You can delete the template by going to your backend > easyblog > posts > post templates and delete it from there. You can also disable the post templates entirely by going to your backend > easyblog > settings > general > composer > post templates and set to "No".
If I'm wrong, kindly elaborate more on your inquiry so we can have a better understanding on your issue. Please advise.