Well, if you see a whole bunch of them in one spot, I would spray them, but those kind of mites mostly live on the chickens. The Red mites are when you have to spray the coop.
I still think you have N Fowl mites... and if it were me, I would Ivermectin all my birds. If you think it is poultry LICE, then you will need to dust or spray the birds with something, as they are apparently not blood suckers. (I have not researched them, only pics so you can identify the problem). This has been a persistent problem for you, which is why I suspect mites.
So no, you don't NEED the spray, unless you have lice or red mites. But I still think it's good to have on hand. For example, I would have soaked that dead rooster to kill off everything, put him in a plastic bag, and stuck it in the garbage. By tossing him, you still have the problem on the property, and the wild birds will bring it all back.