Emergency flasher - bracket support failure

No bracket behind the trim panel. The switch has two tabs that make it snap into the trim panel itself. The stop tabs must be broken off on your switch.

To replace it you remove the trim panel. To do that you remove the three screws from the steering column cover and remove the upper cover. Now remove the two screws in the trim panel at the top edge of the display area. Then gently use a pry tool to remove the trim panel retainers. Disconnect the hazard switch and use a screwdriver to press in the retainer tabs and remove the switch from the front of the trim panel. Snap the new switch into the hole, attach the wiring and reinstall the trim panel and steering column cover.

If you look at the picture of the switch you can see the tabs just below the button that keep it from pushing through the dash. Your initial description sounded like you meant you pushed on the switch and it had pushed through the trim panel and was floating.

If you are pushing the button and that isn't staying in, that is a failure inside the switch itself. There is a U shaped wire with a spring that moves in a Z shaped slot. When you push the button in, the wire shifts into a different spot and holds the button down. When you push to release it the wire moves again and lets the button up.

Not something that can be repaired, instead you replace the switch assembly.

They will tell you the same thing. The switch itself simply snaps into the trim panel. There is nothing else that holds it in place. No other reinforcement behind it to attach to because then you couldn't remove the trim panel itself to replace the switch or the dimmer control.

If someone pushed the switch hard it broke the switch, either the two small sections that keep the switch from being pushed through the panel or the internal switch assembly itself. They are not made for any abuse or even constant use.

Here is the GM factory diagram and switch replacement instructions pulled directly from GM tech system.

Removal:

1 Disable the SIR system
2 Remove the instrument panel (I/P) cluster trim plate from the I/P carrier.
3 Disconnect the dimmer switch electrical connector if required.
4 Disconnect the electrical connector from the hazard warning switch.
5 With a small flat-bladed tool release the retaining tabs for the hazard warning switch.
6 From the back side of the I/P cluster trim plate, push outward on the hazard warning switch. This will remove the switch from the I/P cluster trim plate.

Installation:

1 Align the hazard warning switch to the opening in the I/P cluster trim plate.
2 Push inward on the hazard warning switch until fully seated into I/P cluster trim plate.
3 Connect the electrical connector to the hazard warning switch.
4 Connect the dimmer switch electrical connector if required.
5 Install the I/P cluster trim plate to the I/P carrier.
6 Enable the SIR.

You can do as you wish but you will be wasting money.

The trim panel is just a piece of plastic, there is a hole molded into it that the switch fits through. There is a recessed area that the outer tabs set in (red points toward button) and the inner tab (red lines to the left) simply squeezes into the hole and expand once they are through the plastic.
In the crude image I attached the two red areas right under the button are the only pieces that keep the switch from pushing through the plastic. If they broke off the switch could push father through the hole and as a result the button doesn't push in far enough that the internal toggle can lock it down (on). Instead of ON/OFF you would have push and hold for on.

Nope, the tabs are on the switch. They break off you replace the switch.

The only time I have seen trim broken was when someone smashed it HARD like in an accident or hitting it in a rage, or if they were removing it wrong and snapped it. The switches on the other hand are about like handling fine glass, sneeze hard and they break. For the most part a hazard switch might be used 10-15 times during a normal vehicles life. They don't make them any stronger than they think they need to be.