Scripts also have a last executed timestamp just like automations. Scripts do not have an off switch, because the off switch disables the triggers. Scripts have no triggers - what would turning off mean? But the last thing you mention sends shivers down my spine.
That plus button for automations in the device is a direct invitation to use device triggers, conditions and automations. I avoid them like the plague. Besides that they do not even support the labels and areas you asked about, they are impossible to read - especially if the device is removed. Devices often have multiple entities: cramming all actions for them together is a jumbled mess. If a device breaks down all automations and scripts break with them. You cannot replace a broken device without fixing all references everywhere (which you can no longer read if the device is gone). Scripts not having the button is a blessing in disguise!
LOL: I just checked the available triggers on my Skoda Enyaq device. I can scroll for days through the list 