How long does it usually take for the TW bug team to resolve a bug? I can imagine it’ll depend partly on the severity. If it usually takes a long time, I’ll hire a Linux dev to look at the issue on the off chance they can fix or mitigate it.
Anywhere between a few minutes and never. Sometimes a dev is anticipating a need and is watchful and ready. Sometimes a dev will create a report, then mark it fixed within mere minutes. Most take longer than a day, by a lot.
There is nothing to resolve there. There is no bug yet. Your system stops in emergency mode for some reason. The messages about graphics on the attached screenshot are very unlikely to have anything to do with it. You need to find out why system stops in emergency mode. It almost always is a local configuration issue.
Start with doing what the message before password prompt tells you.