Yep, normal mushrooms should have vulcanising solution, it gives a perfect seal and permanent repair and adds strength back to the hole. You can fit them without and it’ll probably work just fine- the air pressure and the friction from the interference fit will hold it on and the sealant will seal it. But, it’s way more likely to have some sort of failure and it won’t give you the strength that the vulcanised-on mushroom does, so basically don’t, there’s no reason.
TBH with such a small hole I’d just clean it up well and then put a standard bike patch behind it (not a self-adhesive, a proper oldschool one). Even the smallest mushrooms I’ve ever used were designed for a bigger hole, there’s bike ones that start from 3mm. Which is fine, you can make the hole bigger and the mushroom will still give you a completely permanent repair, but, a little interior patch is usually going to be fine, and cheaper.