Using the search function in the map at https://opencellid.org/ , search appears to be working properly for me, though it may not always be obvious.
If there is a cell matching the criteria, the map simply centres the cell, with no further highlighting. If map does not move, either the cell is already centred, or does not yet exist on the map.
You can test to see the behaviour of the map by using search criteria for a cell that you know to be on the map (e.g. MCC 311, MNC 480, LAC 20225, CID 20336641). Move map to somewhere over Europe or South America. Then search for cell mentioned above. You should see map centre over the cell, but no other indication beyond that.
Not sure about the Cell Spy Catcher app as I have yet to try that.
If you would like, you can provide the search criteria of an example of a cell you are trying to find but cannot, and I can try searching for it to see if map is finding it?