Dreadful etiquette by A, and something I've NEVER seen a player do in all the four ball play I've been in. Everybody picks up their marker in A's situation, unless they want to wait until after to practice putting.
That said, I'm wondering why C & D didn't just concede his putt, at which point he can't putt until after the hole is completed. That doesn't excuse A's behavior, of course.
Not disagreeing, But if you are planning on posting a score you should wait until the other team has finished out. That's the problem with our handicap system. In BB you may hit shots and take chances you wouldn't do if you were playing your own ball but you are required to post.
There is a big difference between trying a shot that you might not try if you were playing your own ball because your partner is in great shape, or you don't get a stroke and the other players do, and so on, vs what A did. And yes, while strategic shot decisions might perhaps very slightly skew the handicap results, that's a gate that swings both ways, doesn't it? You also will elect, at times, to play more safely when your partner is out of the hole, or when your opponents are in trouble, or whatever. Over time, in my experience (and I play a LOT of four ball net match play!) all of that washes.
All of that said, none of that has anything to do with A's actions here. This not only was NOT a strategic adjustment because it was better ball net match play, but the USGA provides a guideline for the most likely score if you don't putt out in match play. And even if he's going to putt out for purposes of posting a score, simple consideration of others would say that he waits until the hole is completed before he putts. I've seen many, many times a player with a birdie putt leave his mark and then putt out for handicap purposes when his partner is already in with a birdie, but ALWAYS after everyone else has putted.