@Y2AdminConsultants well you've certainly done a fair bit of testing.
Firstly, is there any reason why you prefer not to go via Pro?
Secondly, the shapefile is an open source file format. Consequently, certain software will author shapefiles with different restrictions to ESRI. For instance, in QGIS you can have field names with spaces between words, which is unheard of in ESRI software.
Without seeing the data, my hunch is that there is a rogue field or geomatery in the point shapefile that's incompatible with the Runtime geodatabase that's created when the map is prepared for offline use in AGOL (if it's a point, then perhaps it's a multipoint shapefile?). However, when you publish from Pro, there's probably some pre-processing being achieved during service publishing.
Consequently, your best bet it to either keep working through Pro, or identify the rogue field/geometry before uploading the shapefile to AGOL.
Another solution would be to copy the data via Pro or load into File GDB and back out to shapefile, to see if that irons out the kink.
David
..Maps with no limits..