It definitely helps in that it becomes another candidate for switching the input, since the Fire TV cube is itself an IR blaster.
In this case, you would probably use an Alexa routine (not a SmartThings routine) to switch inputs. This means you don’t have to buy an additional IR blaster device, but you will have to set up a virtual device in ST in order to trigger the Alexa routine.
FAQ: Can I trigger an Echo Action without Speaking to It?
And again, all that does is change the input, you still need the rest of @fido’s project to trigger from the camera.