Why does SSAS keep restarting every few seconds?
We never found out what was causing the service to stop, but we were able to prevent it from happening. We put a Deny on the Stop permission for SSAS service for the SQL Server (not SSAS) service account user.

So it seems something acting as the SQL Server's service account user was issuing the Stop command for the SSAS account. This was a while ago, and we've since switched servers, and we stopped digging to figure out what was really happening. So it will remain a mystery for the ages.