Question around 'Execute as Role' = 'Administrator`. This doesn't affect the search result produce within the script, correct? Meaning, the search results will still adhere to the User/Role that trigger the script.?

No it will execute the search as if the user role is admin

Interesting. We're not seeing that behaviour. I've started by setting ALL scripts to Execute as Admin, yet, the User/Role that triggers the scripts get script errors indicating that do not have certain permissions

You can't run all scripts as admin. Client scripts always run as the current user AFAIK.

just speculating - I'd think the PERMISSIONS for the searches would use admin, but if you're using relative filters in the search based on employee - like "my team" or "mine" or w/e then I'd guess that would still be respected because that's not a user permission as such.

The searches are being dynamically created and executed at the code level.