Axes are a possibility but you've mentioned speed being important and it's a valid concern.
I wonder if we'll be able to punch them to death?
Axes tended to one of the fastest short range melee weapons that maintained the ability to punch through armor. A short sword would work as well.
There's no such thing as too close a range to fight at for something unarmed. That'd just be risking getting into a grapple against something much stronger. You're also forgetting that spears can be really effing long and still somehow usable, so you could fight a dragon beyond its melee range.
One: It would have to be an awfully strange dragon to be able to grapple you. And actually, unarmed fighting, like boxing, has a larger minimum effective range than knives, since unarmed strikes (for people at least), require a certain amount of momentum and a decent angle to actually do any damage. You can definitely get inside the range of a dragons teeth and claws.
If you're inside of the dragons minimum effective range, it will have difficulty seeing you, striking you with anything approaching lethal force, and dragons aren't built to grapple.
And the point about the spears is semantics - people were clearly talking about spears as a melee weapon.
Anyways, to be honest, you probably would want to START with a spear of some sort, and depending on circumstances, close.
Anyways, the most effective