I think The ECID Scripts get loaded around the same time as Library Loaded...
There is an "opt-in" consent option on ECID (but I haven't actually used that since our team hasn't required that yet), I wonder if that switch would allow the delay of loading.
When I use my Launch Switch extension debug mode on our site, the ECID is fired before my first Library Loaded rule, as well as the analytics scripts added to the page (not yet used, but loaded):
[Experience Cloud ID Service] Created instance using orgId:
[Experience Cloud ID Service] Set variables: {"trackingServer":"xxx","trackingServerSecure":"xxx", etc....
[Experience Cloud ID Service] Obtained Marketing Cloud Visitor Id
[Adobe Analytics] Loading script: //.../AppMeasurement.min.js
[Adobe Analytics] Loading script: //.../AppMeasurement_Module_ActivityMap.min.js
Rule "Library Loader - Rule X" fired.