That's why I don't think he's the one who picks the all transfer targets. It's been reported that at Southampton that Les Reed was the one buying the players. We know that when Koeman took over the squad had just been ripped apart and he was tasked with over seeing a rebuild. They did this very well.
We know that Moshiri wanted a Director of Football for Everton, for long term stability. We know that Stones, Lukaku and maybe even Barkley wanted away for so called bigger clubs. So there was a chance the side was going to be ripped apart. So you need a manager prepared to work under a Director of Football, who has previously done a good job in this league. Koeman ticks all those boxes and more in that he's a big name in the game, in fact Koeman turned the job down until the money got ridiculous. Then you need a Director of Football who can spot good players, up steps the man of the moment Steve Walsh. Who reportedly brought Mahrez and Kante from the streets of France to become Premiership Champions. All the boxes are ticked.
At Southampton they had pace and they had a big target man. He's gone on record loads of times saying he wanted a centre forward and a centre back. In fact even last season he was saying we needed another striker. It hasn't happened. He states frequently about pressing and moving quickly, if you watch him on the line he's always moaning about this. But the players we're bringing in don't fit the system he talks about.
He goes on record multiple times stating he wants a centre forward and a central defender. For me, him going away for the transfer window deadline is a public message that he's not the one bringing these players in. Sure he has some sort of say in it, but he's not buying the players and possibly not even identifying them all.
That's why I don't think sacking Koeman is the answer.