Output Debug-StorageSyncServer -FileSyncErrorsReport to a file

Why is this a bad thing?

Without explaining what your problem (or goal) is you're not likely to get much advice.

Of course, it would be helpful if the product team that supplied the module actually published any information (other than a few examples of "how to use" it).

That said, keep in mind that PowerShell makes heavy use of pipes. If you want the output sent to a file you can pipe the output from that cmdlet to the Out-File cmdlet.

Also, are you sure the output is being sent to the Success stream (a.k.a. SYSOUT)? Perhaps it being sent to the Error or Debug stream? The "*>" redirection operator you tried should have sent everything to a file name that you provided. You can also use "x>&1" (where "x" is the stream number) to send everything to the success stream -- and you can add as many of those redirections as you need (one for each stream if you want) as a way of discovering which stream the cmdlet is actually using.

https://learn.microsoft.com/en-us/powershell/module/microsoft.powershell.core/about/about_redirection?view=powershell-5.1