The elder McMichael was a cop that worked a case with Arbery. He knew who Arbery was, and knew he was a criminal. He also claims to have seen video/images of Arbery inside that house which had had things stolen from it in October, and he recognized Arbery.
Also, Arbery's brother is a criminal, and the cop probably knew him too.
Being a retired cop isn’t relevant...doesn’t give him special privileges to shoot anyone nor chase them down.
As a citizen though the state appears to allow citizen arrest, which those laws will be in the forefornt rather than he was a retired cop.
Further even your post info makes it look like he was out to get the guy.