The "Airspace Control Means - Use" feature class in the "AirC2_ACO_Use.mxd" contains the time fields in format of "string" rather than "date" type. When published as a feature service, and brought into a web map, this time enabled layer does not show. When time is not enabled in the web map, the layer appears.
Esri support articles indicate:
"ArcMap appears to recognize text fields as date for enabling time, but AGOL will not. Once we created a new date field and used field calculator to copy the values over, we were able to enable this without a problem."
New fields need to be created in the feature class that are of "date" format.
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