There were two different types of Falcon rearsight. The earliest versions had a tangent sight similar to those found on the BSA Meteor Mk1 and early post war Milbros dovetailed into the top of the breech block. Later rifles had a spring leaf sight screwed into the block.

I have had better results with a 3x 1960s Diana telescope sight than either BSA or Webley scopes from the same period and you may want to try one of these. I wouldn't recommend rebluing as that would devalue the rifle, which sounds like the rarer early variant from your description. Maybe a tidy up with cold blue if you must?

Kind regards,

John