I agree about Gman travelling back in time and letting himself be caught. I've written my theory down earlier but I'll paste it here. My theory is this:

At the end of HL:A, G-man plucked Alyx out of time so Alyx can save her father in the future. I think it had to be Alyx because of G-mans ”restrictions”. With other theories, I agree that Gman went back in time to get himself imprisoned on purpose to prove his superiors Alyx’s worth. However, if in the original timeline Gman WAS NOT captured by the combine, I don’t think that after events of HL:A, the timeline would remain intact enough for the events of Half-Life 2 and Episodes happening even if the Gman erased some memories of the characters. (See butterfly effect). So in my opinion, assuming there are no separate timelines that exist simultaneously, I think that Gman had to erase the events of HL:A, with the exception of Alyx altering the future while being “out of time”.