We wanted to deploy a FortiWifi with VDOMs, because wireless devices, like smartphones and guest devices, should be completely separated from the rest

We then added this new WiFi VDOM, configured a WAN interface and added a WiFi SSID on it. We then checked on our smartphones, but this SSID did not show up on our devices. However, SSIDs on the root VDOM did (made for diagnostic).

After a long search, we came across this:

If Virtual Domains are enabled, you must select the VDOM to which the built-in access
point belongs. You do this in the CLI.

All commands in this article are to be made in the global config, so first:

config global

The solution was (stated in the document):

# vdom1 is the "new" vdom
config wireless-controller global
    set local-radio-vdom vdom1

If you want to know which SSIDs are really broadcasted, use

diagnose wireless-controller wlac -c vap

Is there really no way to have multiple SSIDs in multiple VDOMs with one (built-in) controller?

In my opinion, the controller should be a global object, while SSIDs belong to a specific VDOM. You then get to this VDOM by connecting to/via this SSID.

Have I missed something?