The scientists did not collect them, it is based on samples and population estimates from that. The 50-100,000 estimate comes from a 2015 Oxford lead symposium
"that in the UK in the order of 50,000-100,000 wildfowl (c. 1.5-3.0% of the wintering population) are likely to die each winter (i.e. during the shooting season) as a direct result of lead poisoning. For migratory swans, this represents a quarter of all recorded deaths. Wildfowl that die outside of the shooting season will be additional, as will those that die of causes exacerbated by lead poisoning. Several hundred thousand wildfowl a year may suffer welfare effects".
oxfordleadsymposium.info
Tens of millions of birds die every year from myriad causes including old age. Where did they all go? We don't often come across most of them as per your long experience.
A 2003 study by the Mammal Society estimated that UK cats kill 275 million prey animals annually, including 55 million birds. How did they arrive at the figure? Did someone count every one or was it an estimate? Should we trust this science?