I really can not add a lot to what has been said here. Many good comments.
The insta buddy in this scenario exhibited no buddy skills what so ever (and pretty darned poor self control afterward too, from what OP says), and when the soon to be lost diver descended beyond the agreed upon max depth for a second time, after being requested by his buddy to ascend, I think the OP did all that was required when he lost sight of that diver.
He descended to what he considered his max, and possibly a bit more, looked for the "buddy", and then surfaced when he was unable to locate him in a reasonable amount of time. The one minute rule has always made sense, unless you have specific knowledge of the site and diver to cause you to adjust that search time a bit.
The description of the lost buddy's actions, post dive say an awful lot about the character of that person, and support the impression of someone too focused on themselves to be a safe dive buddy.
For many years I had no buddies of my own, and traveling, I experience quite a few insta buddies, and have to honestly say I have had a few poor buddies, but never any I felt truly put themselves and myself in any real danger. I have had some who I would not do a second dive with because they simply were not my buddy on the first dive. I don't bother to make excuses, or explain, I just say I am picking another buddy, going solo, or sitting out the dive, if that is my only option.