I just discovered this morning (March 23, 2022) that the newest update to Dashboards (released on March 22, 2022 from the best information I can find) has broken the functionality of embedded surveys. Specifically, non-public facing surveys that require ArcGIS Online org credentials. It seems as though the Dashboard is not passing the logged-in user credentials to the embedded survey, thus restricting access to the survey and causing an error that the user does not have permission to access the survey.
This is causing a significant impact to our organization, so I did file a support request and have a case assigned. I was also informed that the problem has been reported to ESRI already and has been identified as a Bug (BUG-000147847). I have officially escalated the bug, so that's about all I can do at this point.
I didn't see any posts about this when I searched this morning, so I wanted to get something out there for any others who are experiencing this issue. I spent quite a bit of time attempting to identify the problem on my end, thinking it was something that I had wrong on my end. So hopefully this will save someone else the time in doing that.