Thanks all and sorry for the late reply:
I had to learn a bit more before coming back.
What I call the "ignition system" was actually a thermocouple + a thermopile. Because the pilot light was on, I concluded that the thermocouple was doing fine.
The next step was figuring out if the problem came from the thermopile or the gas valve.
I made the bet it was the thermopile, and bought a bunch of tools to verify my guess (a multimeter, some wrenches, etc... yup, I did not own those before...). Turned out I didn't know how to open the fireplace in the first place (this probably sounds crazy to many of you DIY people, but I'm not from the US and had never owned/used/seen a gas fireplace in my life before), I procrastinated, and ultimately my landlord sent some company to have a look.
They opened the fireplace and verified, their verdict is that the gas valve needs to be changed. They'll charge my landlord $200 for that verification which took them 15 minutes, and then they'll charge much more for replacing the valve. Oh well... At least next time I'll know how to open it myself.
The broken glass in the fireplace was either some old decorative element (intentionally broken glass that reflects the fire light), or some glass that got broken before. It had nothing to do with the problem.