Yes, they did have that power. They did not, however, actually use it a lot (at all?) after the Punic wars AFAIK (the senate had in practice taken on the entire

Yes, they did have that power. They did not, however, actually use it a lot (at all?) after the Punic wars AFAIK (the senate had in practice taken on the entire government as a war measure, and effectively shut out the tribal assembly of any decision making), and when the assembly again started to wield that it under the Gracchi brothers, they are basically reinventing the participation of the non-senatorial citizens because the old system had effectively lapsed. And we are by that time in the "it started to break" phase.