Step 1 - check the logs.
I would probably reach out to the server admins (and check the logs) to determine why starting the service as a user other than the service doesn't allow it to start as well. There should be no reason why a server admin shouldn't be allowed to start a service.
Getting a random "access is denied" error though sounds like something is happening on the server that shouldn't be such as the antivirus locking a file used by SSRS, or the SSRS account is getting locked out, or possibly corruption of some sort (disk or memory for example).
But, start by checking the logs. My FIRST step to correcting the issue (after reviewing the logs) would be to clone the server so I can restart things and test stuff without impacting production. Next, I'd log in as a server admin, try to restart the SSRS and wait for it to fail. Then I'd check the logs, correct the issue, and try to restart SSRS again. Repeat until SSRS starts successfully.
Next I'd check the AD (you are using AD, right?) logs to see if the service account had been locked out recently and if so, from where. If that came back empty, I'd be running a memory check tool on the server as well as chkdsk on the drive.
Might not hurt to check that you are fully patched too. Could be a known bug that is fixed in a future patch.
But step 1 - check the logs. Check all of the logs that make sense. Windows, SSRS, AD, etc.