Getting back to Hawfinches . . .
The best habitat for them is very high species diversity open woodland - i.e., a botanical garden or an arboretum. They like diversity of food sources more than any single species (BWP lists 57 different genera of trees & shrubs used for food!).
There's actually no evidence that Hawfinches ever bred in Britain at all before the start of extensive exotic tree planting toward the end of the 18th century. British native woodlands just don't have the species diversity they require.
Michael