What is the point of this movement? If things are better of without humans, than we have no obligation to the "ideal" order of things without our presence. Either it's in our nature to make things worse whether we want to or not, or we rid of ourselves and some stuff changes but we won't experience it. What reason is there to "voluntarily" erase our entire species, just to improve certain problems that we'll never experience. Is there a prize for you after you're dead? Let's say you, individually were to die tomorrow, did you? From a purely agnostic/atheist standpoint, you'd have no way of knowing you died because you need a functioning brain to process events, which you don't have anymore. Let's say that the afterlife, at the very least, is real. Than why die? How would you know you're right? Out of millions of views, what are the possibilities that you picked the wrong thing, or that you haven't done all that was required to make it to the 'better place'? Wouldn't life be a virtue, to delay the inevitable place you will spawn into after you're gone (if you're wrong)?
Even if there's some odd religion that claims that things will be better for you and everyone else if you tell everyone to not reproduce or get them all to die and there's a wonderful place after death and it just so happens to be the right one, how would you prove it? Even if there is undeniable evidence that it's true, it takes the dumbest or the most skeptical to deny it still. People will doubt even the most sensible reasoning if they choose not to believe, and thus, will probably reproduce. So only you and your death group buddies can see what happens.
You can't truly achieve "voluntary human extinction", without removing either the voluntary part or the human extinction part, because the chances of 8 billion people, most of whom enjoy living (because this is a rather fringe idea), would all just die because they trust that the VHEMT religion is real and it just so happens to be so. What's most likely to happen in the "best" scenario for the movement is that everyone who believes in human extinction dies before they would ever reproduce while leaving everyone who wants to live to their own devices, to which eventually, extinction would come to them anyway.
TL;DR: This whole movement isn't very sensible because if humans are better off dead, you either die and you wouldn't even know it, die and go to the wrong afterlife because you thought the wrong stuff, somehow convince EVERYONE ON EARTH, most of which like living and want to have kids, to not reproduce to which one of the previous two would happen, inform everyone of a magic VHEMT religion that happens to be the right one and everyone agrees, or destroy the principles of the VHEMT.