I am experiencing a similar issue with Experience Builders Business Analyst widget. I can draw a polygon or search successfully, but when I draw a point on a map, it usually doesn't draw until 4-5 attempts, if, at all. I have tried waiting to draw a point until all of the map layers load and it makes no difference. I also attempted to draw a point on the map at various map extents, which also does not appear to make a difference. ESRI really needs to get this widget's basic functionality working, or remove it. I am working in ArcGIS Online.
I was experiencing a similar issue with layers and polygon working, but not the point. In my case even though I added the AGO locator for the search, the widget was using the locators that were listed under my organizational utilities in arcgis online. In my case, the first locator was no longer available, so I updated the url to a currrent geocoder and once that was updated, the point tool worked. It doesn't seem clear that the widget is using the locator listed in the organizational utilities, so it took a bit to track that down. I suspect if you don't have locators listed there, you won't have the issue.
