I've just run into the same issue and was very happy to find the above workaround. I too was trying to push 1400 features into a feature service without any luck, with the same error 'Could not access any server machines' which is normally only seen when your entire ArcGIS Enterprise deployment is having issues. Decreasing the 'Features per request' parameter from the default 1000 to 400 did the trick.

My details:

ArcGIS Enterprise 10.7.1 (feature service running on ArcGIS Server with the data copied to server).

FME Desktop 2021.1 (the issue also occurred on FME Server 2021.1).

Is anyone in the FME community successfully writing daily changes to ArcGIS Enterprise Portal hosted feature layers from an ArcGIS service endpoint layer? I'm also getting the error 'Could not access any server machines. Please contact your system administrator.'. Details: 'None'. Adjusting the Features Per Request Parameter doesn't help. Workbench version 2022.1

@Gary Christensen​ That error you are experiencing is often a result when your entire ArcGIS Enterprise deployment is having issues, e.g. not being turned on or not having the correct permissions to the feature layer. If you contact your system administrator and are still running into issues, you may have more luck posting a new question with your log and more information on your connection.

I'm glad I found this reply to this issue! I've been struggling with updating one layer, but not all, in one feature service, with the 'Could not access any server machines' error message. Once I reduced the 'Features per request' setting the error went away. The problematic layer consist of multi-line features with many parts.

The suggestion from @ngoorman​ is a good one and helped. I also found if I kept getting this error reducing the number of vertices in the polygons going to the ArcGIS Portal writer with the Generalizer helps.