But the law is on their side.
Just hypothetically, how would this play out, in a modern setting, if both sides were intransigent? Short of physical violence to the monarch and all heirs, what legal way do "the subjects" have out of the current situation if the monarch doesn't want to cede power, and refuses to approve any laws attempting to change things?
If 100% of the citizenry is against the law, the law has already lost.
As far as how it would play out, IMO we have two historical examples in the French Revolution and the Magna Carta. If it's just the lower classes and nobody will listen, you get violence and the power of the monarch is slowly eroded to the point where they can be eliminated without causing any additional uproar. If it's everybody, the monarch is essentially getting told "sign this or remain a figurehead with reduced powers, or you're going to be in a forced retirement". Heck, iirc that's kind of how even the Russian revolution was going until the White armies became dangerously close to freeing the Tzar and restoring his power... though in that case they were just itching for a chance to kill him anyway