I don't know if anyone else has found a solution to this problem, but if you have, please share it! I spent several hours with Esri tech support last week renaming files on my C drive, trying all sorts of different configurations in options, etc., and ultimately doing a new clean install of ArcGIS Pro 2.6 under their guidance, but I still get the dreaded "Indexing is in progress. Please wait or search again later." Argh! For now my workaround is to use the old Desktop version of Catalog (10.8), but it sure would be nice to search right within Pro projects.
For the record, the support technicians I worked with were very pleasant and as helpful as they could be, but clearly this has not been documented as a common issue as they could not offer a solution. If I find or hear of a solution outside of this thread, I will be sure to share it.
Thanks!