No, you cannot do that. This is not part of an email template or a template override, it's how the Akeeba Ticket System – Email plugin works.
There is a solution to that, though.
In your hosting control panel create a mail forwarder. Set it so that it forwards to the email addresses of all of your support staff. Create a Joomla user with the email address of the forwarder and assign it support staff privileges. Assigning a ticket to this user will send an email to the forwarder which will forward the email to all support staff.
That said and seeing all your tickets I don't think that what you are looking for is a support ticket system. You are looking for a hybrid between a forum, a ticket system and a CRM. There's no off the shelf software or service that can be reasonably customised to your specifications. You actually need someone to create bespoke software for you. We cannot do that for you, nor can we make ATS work according to your specifications which are far removed from the operating principles described in the documentation.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!