I've been wondering how to do this for a while too, just came across this thread.
Kristian G, the trick you proposed is interesting, not sure what version you got it to work on but in AC19 I can only get it to work if I have no criteria in the schedules selection, which doesn't make it particularly useful.
But it does highlight that scheduling labels is doable, I switched over to a text object a while ago for most noting on plans as it can display across stories, be scripted to have minor alterations to widths, positions and detail across scales, be controlled with MVO's and be SCHEDULED!
Unfortunately you can't link it to anything *coughs* Labels *coughs, and the schedules don't find instances of the note in elevations, sections, details ect.
It just makes sense for auditing, I have a whole bunch of schedules setup to help me identify & correct errors in object and element parameters, would love to be able to do it with text and labels too!
Hope they get it to us soon, or someone more skilled than I figures out an add-on to bridge the gap...