It didn't have to be done in public and by the manager. Indeed, it definitely shouldn't have.
The club has every right to protect its assets and sell them at peak value when they want to. But it should be done at boardroom level and not by alienating the player in the press.
Once that deadline went down by Koeman in the press, if Barkley signed it he'd have been publicly bending over and people would have called him thick and played like a fiddle. In essence, that ultimatum ensured this drags out at the bare minimum, because no agent worth their salt would have done a damn thing once Koeman said what he said, or at worst it publicised our weakness on the issue and devalued the asset.
If Barkley signs now, it's after Koeman's deadline and makes the manager look foolish. If we sell at a stupid price, it's because the manager basically transfer listed him and exposed our weakness, making him look foolish. He's done anything but 'play a blinder'.