Turns out the last update wasn't the last before release.
In this patch .v1.0.81226, the Main Menu no longer pauses the game. This hopefully fixes the issues for some people who couldn't start the game properly.
Please report if you have any issues.
Hotfix .v1.0.81226
Source: Steam Community (opens in a new tab)
Build and release context
Recorded Steam release date:
Within one calendar day of this post (UTC). The listed date does not establish an exact launch time.