I would like the map to default to showing the current location (and maybe past ones) in addition to my home or current location.

If the main issue is too many carriers, then you might be able to use the favorite carrier list to prioritize them.  For example, check for a match from one of the favorites first, then continue to the rest of them if it wasn't found.

However, even if it's not 100%, suggesting a carrier based on the tracking number would be a great help.