Yes and no. Up until one of the recent canaries, the logic was to keep warnings and errors until the user clicked “dismiss” on them. However, we recently changed that so the status will be cleared after each run, meaning that the icon will now reflect the last backup run.
The downside to this is that it may be possible to miss warnings, but the logic is that if they are transient, there is no value in keeping the highly visible. You can still view the logs if you want to examine the backup history.
If you use the latest canary or experimental, the icon will change as you expect.