No, you shouldn’t need to be online unless you’re building the site with a version of PHP, database, or web server that Local needs to download.
Looking at your error logs, I believe I found the issue:
{“thread”:“main”,“level”:“error”,“stack”:“Error: Unable to read hosts file at C:\WINDOWS\System32\drivers\etc\hosts\n at HostsFileService.shouldUpdateHosts (%%appPath%%\main\dns\hostsFileService.js:1:1721)\n at HostsFileService. (%%appPath%%\main\dns\hostsFileService.js:1:1250)\n at HostsFileService.flush (%%appPath%%\node_modules\debounce-promise\dist\index.js:54:74)\n at listOnTimeout (node:internal/timers:559:17)\n at process.processTimers (node:internal/timers:502:7)”,“message”:“Unable to read hosts file at C:\WINDOWS\System32\drivers\etc\hosts”,“timestamp”:“2023-03-15T08:47:27.568Z”}
Local isn’t able to read (or update, I assume), the etc\hosts file located at the file path above. This is usually due to antivirus or even Windows Defender preventing Local from making changes on the machine.
Check out this post for some steps to resolve the issue - Help! I'm running into an EPERM error with the hosts file on Windows
Another option is to switch off of Site Domain mode and use Localhost routing mode instead. This will also resolve the problem.