In the context of Linux, if one obtains root via a privilege escalation exploit, they may deploy a rootkit via AMT or make other system modifications that one won't be able to evidence/remove by trivial OS fixes.
The scenario is:
An attacker gains remote code execution within the context of your web server (e.g. www-data or a user Apache runs at, or pick your favorite web server). At this point you should be able to identify these events in your logs and figure out how did they access the system, and potentially recover/patch/remove backdoors.
Via a privilege escalation vulnerability, they gain root/UID=0 privilege. Still, logs may be available, and you may be able to patch/recover.
They exploit the AMT/IMT vulnerability and gain AMT execution. At this point you lose any regular visibility to as what have they done at AMT level, and they may use this access to install a backdoor or rootkit that may be really difficult to discover/detect or eradicate.
The temporal score for you may not be the same as a unauthenticated RCE, depending on the context of the system, so I'd say it is a medium criticality vulnerability.
Case #3 applies also to authenticated attackers that may want to gain persistent access or mask their activities in a way that would not be visible to you via a SIEM/remote syslog or other means.
I hope this helps clear the issue.