You're correct, it is indeed contradictory. Taken purely logically, then, if neither can live while the other survives, then if Harry is alive Voldemort must be dead and if Voldemort is alive Harry must be dead. Since we know both Harry and Voldemort are alive, the statement is clearly false. Since the statement is part of a piece that refers to them as alive, and one killing the other, then the passage as a whole is clearly illogical.
However, there are a few sensible ways to read it.
One is to suggest that their life under this doom (in both the common sense, and the original sense of being fated to something) is incomplete. Neither truly live until after it comes to pass.
The other, is that as a prophesy it is speaking about the future, so even though the present tense is used, it should be thought about in terms of the future. This is an unusual use of tenses for most cases, but reasonable if we consider that we're talking about a magical trance that leads people to speak dialogue in a completely different register and rhythm than that the author normally uses! And this also makes it make perfect sense; some point in the future will come, in which one of them must be dead in order for the other one to survive.
Spoiler: Hover over the text to read it if you have read the end of the last book, or don't care about the plot being given away:
It also has a different interpretation, that becomes clear later. Voldemort is in a state that is neither life nor death, and for that reason cannot be killed. This state can only be ended when certain objects are destroyed, and Harry is one of those objects. This means that while Harry survives, Voldemort cannot truly live, and cannot truly die. Harry dies, and comes back to life. Harry's death is the destruction of the last object that keeps Voldemort in his non-life/non-death state, so Voldemort truly lives when Harry stops surviving. Then they can kill him.
The strangeness is justified as fitting with the general strangeness and cryptic opacity of the prophesy as a whole (again, especially in taking how it doesn't fit with the normal style of the books). It's justified further in light of the plot twist above.
It's also deliberately strange in its phrasing, because you're meant to be wondering about it until you've read on through the series.
You can paraphrase the proverb as:
Where there does not lack will, a way opens.
or:
Where there is no lack of will, a way opens.
Both want and not call for a little explanation.
want
The verb want can mean “lack or be short of something desirable or essential”.¹ This is an archaic sense of want, and Tolkien was a language expert and fond of archaisms. You see this sense of want in sentences like
Where obedyence wanteth … there is no goodnes. (The Pilgrimage of Perfection, 1531)
Where Power wanteth Rigor weakneth. (The Illustrated Magazine, 1863, quoting Doctor Dee, 1588)
The Lord is my shepherd; I shall not want. (Psalms 23:1 KJV, 1601)
If you marry me, you will want for nothing.
not
Tolkien’s syntax “will wants not” (N V not) is an example of simple negation. Again, Tolkien is using something rare in modern English, but this syntax was once common. You see this syntax in sentences like
Jesus saith unto her, Touch me not. (John 20:17 KJV)
Waste not, want not. (proverb)
She loves me … she loves me not. (children’s game)
Simple negation is also discussed in a related question.
Best Answer
The original was actually Shakespeare: all that glisters is not gold, but that needn't concern us here.
OP has simply misparsed the sentence - it actually means "Not everything that is gold glitters" (which is to say, "There are some things which are gold that don't glitter").
You can always Google "every x is not y" for more discussion of why this type of construction should be treated with caution. As it happens, I already knew what it means in this particular case (and I knew it was originally glisters), but I think the bottom line is the statement is inherently ambiguous, so you have to go for the interpretation that makes most sense in context.
Tolkien experimented with several variants of the "quirky inversion" of Shakespeare's original before finally settling on the The Riddle of Strider version (that appears twice in The Fellowship of the Ring). But I quite like this somewhat more "pithy" earlier draft...
(I don't know whether the punctuation/capitalisation there was actually what Tolkien wrote).