As @moreamazingnick already wrote, we don’t use the Icinga GmbH’s PowerShell service because we don’t use the PowerShell checks.
what service does the linuxfabrik use then?),
None, the check plugins get called by the Icinga2 Agent service with parameters and the exit code and stdout are the return channel - same as on Linux. No separate icingapowershell service required 
Does the problematic host now communicate with the icinga master?
I thought the script itself was downloading the linuxfabrik-monitoring-plugins.msi.
I didn’t put in the linuxfabrik-monitoring-plugins.msi download and extraction into the script as our server team is temporarily attaching a share containing the script and msi while new servers gets provisioned.
If you want to download the msi in the script, it should be a quick search to figure out how to or just ask an AI to give you a PowerShell code snipped that downloads a .zip and extracts it.
If you’re not shure about what a script does, I advice you to post scripts like this into an AI and ask it to explain, what the script does, to avoid security risks. I know I’m a person with a reputation on here but it’s still better to never trust a script and examine it’s content before running it!