First off, Koeman has proven himself neither a winner or loser yet.
Second (and again) this is not a questioning of the right of a manager to manage, just how he gets the message across and who to is the issue.
Third, yes, I agree that this may well be a case of Barkley being viewed as surplus to requirements and the softening up sale process is underway.
On that last point though: Koeman says he's a player who can fill the #10 role (personally I think he's wrong. He isn't blessed with the quick feet and thinking you need for that role), but he can also play one of the two CM places. He's right on that, so try him in there before letting rip. I'm not sure that Koeman really believes Barkley can do the role though. He's like most managers of non-elite clubs: he likes to go belt and braces with two that will get a foot in and/or screen the back four. Barkley cant do either, imo. He can play further back in a team that is so good everywhere else that they can afford to go with one out and out destroyer and leave the other CM to be there to mop up and drive forward with the ball.
So, basically, I think it boils down to this (in private or public) Koeman's message to Barkley is to adapt to the requirement of a non-elite club in that area of the park or it my be best if we sell you on to a team that can use your more specialised skills.