Thanks for the contribution @petervandivier. throw should be a terminating error, and ErrorActionPreference is supposed to only allow you to alter the behavior of non-terminating errors. I wouldn't expect you to be able to bypass the throw with SilentlyContinue, and yet I was able to repro exactly that.

-ErrorAction documentation (ephasis mine):

Determines how the cmdlet responds to a non-terminating error from the command. This parameter works only when the command generates a non-terminating error, such as those from the Write-Error cmdlet.
...
The ErrorAction parameter has no effect on terminating errors (such as missing data, parameters that aren't valid, or insufficient permissions) that prevent a command from completing successfully.

There are a number of places where this could additionally have issues (like for example in the very next method: Get-GitHubTeamMember):

I don't want to fix these all one-off. I'd like to address all of these in a single changelist so that we can ensure that we're approaching that issue in a consistent manner across the entire module. And I wouldn't want to do that until I had a better understanding of why -ErrorAction SilentlyContinue is altering the behavior of a terminating error.

If you have more context as to why PowerShell is allowing -ErrorAction SilentlyContinue to bypass a terminating throw command, please add it to the issue that you opened so that we can come up with a holistic approach to fixing this issue across the entire project.

I've updated the original issue with more details on the problem and suggested approaches that can be taken to move forward.