Which is what I said, apart from the fact that you are incorrect in your assumption that they are only recognisable as a blob of feathers. Most of those on the runway have clear collision damage but are, however, recognisable. I have been lifting birds from runways for many years and can tell the difference between vortex damage and collision damage. The main difficulty is picking up pieces after the bird has had a trip through an engine or the occasional prop A/C chopping something up.
All suspected bird strikes are reported in the UK even if they cannot be matched to any particular A/C at the time. Quite often it's not until they reach destination that any evidence is found on the body work or engines. That is why both Airfield ops and Pilots fill out mandatory bird strike forms.
however, in most cases the bird on the runway is found after a reported strike and can be matched to the flight reporting it.