Perhaps I was being too coy...
Critique of what you've got (qualis vir est qui mundum meliorem non reddat): literally 'what sort of man wouldn't make the world better', or 'what sort of man doesn't make the world better'. I read that over and over, and it keeps telling me the point is: it is inevitable that men make the world a better place. As if I asked, 'what sort of man doesn't love his children?', meaning you to understand that love of one's children is a natural, inevitable part of being a man, and you'd have to be weird or inhuman not to do it. That's not what the original sentence is getting at.
The problem the English poses is that it emphasises the second 'man' -- that's why it says 'man' twice, so that the second time you get that it has a special meaning, 'real man, manly man' as opposed to 'human being'. Sure, we've got a word in Latin for a 'man as opposed to woman', but since it was such a male-oriented culture, it can be used loosely for 'person'. So the word itself isn't going to give you any help: when you say 'quis vir' or 'qui vir', it could either mean 'what man, and I really do mean man...?' or just 'what person?', i.e. little more than 'who?'.
Some of the translations in your original post just bury their head in the sand. In others, there is an obvious attempt to deal with the problem. Possibly 'nemo vir est' is the most successful because it looks so odd -- i.e. it won't allow you to read it the soft way, as 'there is no man who...'. I'm inclined to write 'quin' instead of qui, in which case there's no argument about reddit/reddat:
nemo vir est quin mundum reddat meliorem. = no-one is a man but that he renders the world better. But I'm even more inclined to write 'nisi' (unless) or even 'nisi qui' (except who), since the sentence means effectively 'no-one is a manly man unless he makes the world better'
'qualis vir est...' seems a bit desperate to me: it acknowledges the problem, that we're not talking about 'person' but a certain quality of manliness. But it doesn't solve it.
To answer the reddit/reddat question (if you've read this far, you deserve a straight answer): if I said 'he is the sort of man to make the world better', I would use the subjunctive talis est qui mundum reddat meliorem. So putting it in question form 'what sort of man...', I would write 'qualis est qui... reddat'. If I put 'vir' in, it would alter the sense hardly at all, so I guess I would keep reddat:
qualis est vir qui mundum meliorem non reddat.
Anyway I tried to solve the 'man' problem head on. I added a few words, but what I gave you would be understood in the correct sense.
Though slightly miffed at your response, I acknowledge a small apology is in order: it's in the nature of this kind of exercise that we come up with many alternatives that we often argue about. Unfortunately, this wrangling is of little interest to you! But I hope you'd prefer to get good Latin than so-so Latin.