It wasn't just Miss Finch, it was the entire board of The Feathered Friends of America: their purpose was to place "stray birds" with loving bird families. Big Bird, who lives all by himself, at the age of six, was seen as being a stray bird, who needed to be adopted into a loving bird family. Miss Finch knew the Dodos were interested in adopting, so she took it upon herself to see that both the needs of Big Bird (who seemingly needed a family), and the Dodos (who wanted to adopt) were met.
Being a bird, Big Bird did think it would be a wonderful idea to have his own bird family, but after spending some time with the Dodos, he sees that the dynamics of the family were not what he was expecting, and it really takes the cake when they feel he shouldn't be best friends with Snuffy because he's not a bird. As for why Miss Finch was so determined to get Big Bird back to the Dodos, there was really no real relevance to that, as this is a kids movie, and really, it was only a plot device to give the story conflict. In a real-life scenario, who knows... maybe if the Feathered Friends saw Big Bird ran away, her job would've been in jeopardy since she took responsibility for placing Big Bird in a proper bird family. But again, that's not specified, and that very well may not even be the case... again, it's mostly for conflict in the story.