I could imagine a super low power coprocessor that's always on would be a real useful tool. It could check my email and other low power tasks.

And remote management isn't bad if it's entirely under my control. It's the closed nature that makes me distrust it.

This has already been solved. Modern devices come with low power general purpose cores and OSs can wake up briefly to check for new messages. I just can’t see why you would ever want to remote manage your own laptop where some software installed on the OS isn’t sufficient.

Intel recent CPUs come with efficiency cores and support S0 standby, aka "Modern Standby", which can periodically wake up and do stuff like check for new emails.

There have been earlier attempts at achieving this functionality by Intel, can't recall what they called it right now, but it has existed for a while.

They do but it's not nearly as low-power nor as constantly available as Intel ME is.