Any idea why OpenAI cared about this name removed from the paper?
They didn't want his name removed from any paper. The invitation was to write a new joint paper between Buckmaster and OpenAI. An invitation Alpoge couldn't accept and OpenAI wouldn't make since he worked for a competitor.
He works for Anthropic nowadays.
Because he works for Anthropic. Supposedly this project was not part of his official capacity as an employee of theirs.
However, if they had published first, it's hard to imagine Anthropic not taking the opportunity to claim "our employee solved this Millennium prize problem using our AI".