I wonder if some of the impatience with Koeman is the fact that it seems to be a short-term stopgap. Everything about his demeanor as well as the general assumptions around his move from Saints makes it seem like he's been brought in to steady the ship and get us on an upward trajectory for the real future manager to come in.
Thing is, when you're a short term solution, people expect quick results. When you don't layout a vision for where you plan to be, the expectation is that you hit the ground running with results.
I have no idea if Koeman actually sees us as a stepping stone, or if we see him the same way. But the appearance is that we see each other as a means to an end. It doesn't really matter if that's the case from a pressure standpoint.
Martinez got a little more patience because he clearly had a plan (it was not a good plan, as it happens), and then got a LOT more patience due to that first season. It's no surprise that people are turning on a 'stopgap' manager that isn't getting results quickly.