I repeated your workflow of a feature class with empty default field type of Date. Published the feature service to ArcGIS.com, make a webmap with this feature service, open it in ArcGIS Collector on my Galaxy Nexus (Android 4.2.1) phone. Collect a new point, tap on the Date field, the date picker is the Android system's date picker with today's date shown.
Could you please tell us what phone you are using? Android version? If possible, could you please share your webmap URL. I will look into it further.