The reasoning behind having games end at set times, rather than requiring one player wipe out all others, is that Vox is largely a diplomatic game. A requirement that one player defeat all others would render joint victories impossible and place strain on alliances. Also, as empires grow they become increasingly difficult to manage. A requirement that a single player control the entire board would drain much of the fun from the late game and fill it with tedious combat.
Additionally, since your only goal is to have the greatest score when time runs out, you can bypass your opponent's strong points and concentrate instead on their weaknesses.
After a game ends you may view it for a time under the "Recently Finished Games" heading on the Game List. You may also view them from the 'History' tab of Your Account. After three months, games are archived and removed from the database.