Yes, this is a known pain point. Spine is designed to work offline after the initial setup, but there are some situations where it still tries to reach the server on startup and fails to launch if it can't.
A few things worth trying:
Specify the server via command line
Sometimes the launcher has trouble resolving the default server. You can try forcing it to use a specific regional server. From Command Prompt:
"C:\Program Files\Spine\Spine.exe" --server us
or try eu or jp instead of us.
Test latency / ping
You can also run:
"C:\Program Files\Spine\Spine.exe" --ping
This tests latency to each server and may help diagnose which endpoint is failing.
Use a proxy if you have one available
If you have any kind of proxy available (even via a mobile hotspot routed through a proxy), you can specify it:
"C:\Program Files\Spine\Spine.exe" --proxy server:1234
See the full proxy documentation here: https://esotericsoftware.com/spine-proxy-server
Check your spine.log
The log file will have more details about exactly what's failing. You can find it at:
Windows: <user home folder>\Spine\spine.log
You can open that folder directly from Spine's Settings > Files > Log button (when you do have connectivity).
Regarding your feedback
Your suggestion about a realistic offline grace period (days or weeks rather than requiring a live connection) is very reasonable feedback, especially for situations like broadband outages or travel. The current behavior where it fails entirely rather than falling back to cached/offline mode is something worth flagging — the expectation from the documentation is that after initial install and activation, Spine should run offline, so the fact that you're hitting connection errors on startup suggests something in the launcher's startup check isn't gracefully falling back as intended.
Reinstalling with the latest launcher from your license page may also help, as that's resolved similar issues for others in the past.