Try relogging and see if it works. Even if you do manage to hit the start button you still might or probably be unable to get the na/eu server list.
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Same here with the disconnect from lobby problems and everything... Hope they upgrade the servers soon and at least a new update about the situation today.
On Lodestone, whenever one of your friends logs in, it bothers to post on your message wall "[so and so] has logged in". Why would they do this, unless logging in succesfully was itself considered an achievement, and a part of the game?
Therefore, failing or suceeding to log in is considered a major element of the gameplay (and by all accounts is still less tedious than actually playing the game pre-ARR)