This issue is addressed in ArcGIS Pro 3.2.
It is documented here that searching for server connection files is indexed and supported, but not the contents of a server. See the note here: https://pro.arcgis.com/en/pro-app/latest/help/projects/index-project-items.htm#ESRI_SECTION1_6F022A465C2D4471AC03B0D831AC0592.
Therefore, the behavior that is described as not working is ArcGIS Pro working as designed. However, some changes made to ArcGIS Pro starting in ArcGIS Pro 2.7 changed the tooltips that are displayed in the search control to reflect the name of the current container. Some items are containers where the tooltip there implies a search would be successful when it actually cannot be successful. The control and the information provided to account for scenarios where the search is not expected to work are to be improved.