Hi Robert,
thanks for the reply. Hosted Layer Views are new thing to me (and for the rest of the department), and sure looks promissing. However, it seems to only fix half of the issues. It does help with hiding those fields that aren't needed by the general editors for the layer. But arranging order of the fields in Map Viewer still doesn't save. Not even within same instance. I tested this by changing order of two columns, closing Attribute Table and saving the map. Opening Attribute Table again, the order is switched back to default. Changing order for Pop-up configuration doesn't work in Map Viewer either, seems like that is only effective for Classic.
If the order of the fields could be changed to different default directly from layer Data settings page, that probably would be best. Similar to how visualization default is done from layer Visualization options, not from Map view. Though it seems like bug that Map Viewer Attribute table forgets user set order when table is closed and reopened.