however, in most cases the bird on the runway is found after a reported strike and can be matched to the flight reporting it.
How do you know this for a fact and not an assumption.
For multiple birds strikes in the same event it's not an issue. However for the the random single bird strike, there is a presumption that what they find must have been from the aircraft reporting it. My data says that the bird indentification reports are wrong 50% of the time.