As per Microsoft supported option, you can follow below article and use site definitions instead of site template.
"saving publishing enabled sites as a template is not supported and it does not work. You cannot use this option to create new sites. Instead you should create site definitions (..\web server extensions\12\template\SiteTemplates) and provision ...
According to my understanding you are looking for Item-Level Permissions.
Below is a good article on how to enable item level permissions in SharePoint. maybe it will help you with your requirements.
How to enable Item Level Permissions in SharePoint.
You have 2 options:
A. Use custom permission (using some wf/flow you could trigger it automatically upon creation and grant only the edit rights) but having list with custom permission has its performance drawbacks. ...