You are a bigger club, and can you can offer him things that Saints can't, but that doesn't change the fact you have to overhaul your squad, which is not a easy thing to do, even with a lot of money, look at Liverpool, United and Newcastle.
In pure footballing terms, I'd argue that the Saints squad is stronger than Everton's at the moment. After your summer of investment this might change, but Koeman is taking a risk saying that I am the man for the big overhaul, it's high risk high reward, and you can understand why some people would see it as a risk not worth taking for Koeman.