Hey @oligraph –
That sounds like something isn’t finishing the installation of WordPress correctly. Can you please provide your local-by-flywheel.log file? See this Community Forum post:
How do I retrieve Local's log file? - FAQ - Local Community for instructions on how to do so.
To your specific question of if you can download WordPress manually – yes, “sort of”. If the various pieces of the Local site’s server software haven’t been configured correctly then there isn’t any way to download and install WordPress manually.
Taking a look at the Local log should give additional info about what sorts of things are preventing the site from working!