Eccup Reservoir, thank you for the reply.
Based on the documents below, only security and important/critical updates are delivered using the ESU update program. Also, delivery, download, and application of ESUs for Windows Server is no different than other Windows Updates - so nothing extra needs to be done.
Download and installation of Extended Security Updates What is the Extended Security Update (ESU) program?
Furthermore, for .NET also, there are support available for Windows Server 2008 R2 ESU and Windows 7 ESU which include .NET Framework 4.6.2 thru .NET Framework 4.8 and .NET Framework 3.5 SP1 only.Azure Update Manager retrieves the assessment information about status of system updates for it specified by the Windows Update client. For more details, see the information available here. If you do not see newer updates getting listed as available, it would mean that there are no applicable updates as recognized by Windows. You have also confirmed the same by stating that you don't see pending updates even when logged directly onto the target server.
I would suggest reviewing some of the target servers to verify that they indeed miss the critical .NET framework updates as per the supported scenarios. These can be installed directly on the machines as well, if identified, as only security and critical updates are delivered by AUM. Hope this helps.