Hello Arnou,
Sorry for the delay in getting back to you, I wanted to try running this simulation so I can provide some direct feedback about the steps to take to set this model up and get it running with Event Simulation.
This is a great model for the Event Simulation study type. Your intuition is on the right track, you only need gravity and contact in order to get this model to work properly. The general idea is that you'll expose the model to gravity, run it for a short duration and see if the door is starting to fall off the rack. You could run this model with a time duration of a few seconds and watch the door fall towards the ground, but that would be overkill since you only need to know if it will stay up or not. For this purpose, I would start with a time duration on the order of a few milliseconds (maybe as many as 10) to see if the information you need is clear by this point in time.
The first thing I would recommend doing is simplifying this model quite a bit. There's a lot of detail in here that doesn't need to be included in order to get the answer you're after. For example, the complicated support system that holds the stand upright is not of importance for this simulation. We'll assume that you've designed and simulated the support structure properly, so that when a door is placed on it the weight of the door will hardly move the stand. If that's the case, we can remove everything from the model that isn't directly supporting the door and simply fix their locations. I took your model and simplified it down to just the rollers and the supports that hold up the back of the door. This is what it looks like:

Since we also don't care about the structural response of the rollers and the support brackets, we can make them rigid bodies in the Event Simulation study. Then we can apply a structural constraint (fixing Ux, Uy, Uz, Rx, Ry and Rz) to each rigid body to fix them in place. Here is the simplified support structure with rigid bodies and constraints applied:

Gravity is enabled by default in Event Simulation, so the only thing left to do is to enable contact with friction. This is incredibly simple to do with Event Simulation. Contact modeling is completely general-purpose and automatic, so all you need to do is turn it on and specify the coefficient of friction you want to use. If you click the "Global Contacts" option, the only thing you need to modify in the dialog is the coefficient of friction. I did a quick online search for a common friction coefficient between wood and smooth metal and chose a number of 0.2. I recommend doing some more research to determine what a good number for the friction coefficient is for the materials you are using.

We're all done, so now we are ready to drop the door and see what happens. I set up the Event Simulation to run for 10ms (0.01 seconds) to see what happens once the door comes into contact with the rollers. With this short of a duration there won't be enough time for the door to literally fall to the ground, but there should be enough motion to infer whether the door is going to stay put. Here is an animation with a view from the side of the door showing the motion of the door at the interface with the rollers. I included a surface probe at the bottom corner so you can see that the X-displacement is actually growing here.
You can see that the door is actually sliding out as opposed to falling forward. Here is a plot of displacement in the X-direction showing the whole door.

The dark blue contour color corresponds to a negative X displacement, which shows that the top of the door is actually falling backwards. If we magnify the displacements a little bit to better see what's going on, this confirms that suspicion:

From this it's clear you need supports holding the door that are higher up to prevent the backwards rotation. After you do that then you can try again to see if the door is going to fall forward or stay put on the rollers.
For reference, this simulation took about 40 minutes to run. I've attached the Fusion .f3d file in case you'd like to check it out and run it for yourself.
I hope this is helpful to get you started doing more of these types of simulation. Let me know if you have any questions about this process, I'm happy to help.
Jeremy Wiesner
Research Engineer, Fusion 360 Event Simulation