All that, yes. But they don't work too well if the battery runs out halfway down the straight... and there is too little charging.
I wasn't politicizing E vs ICE... but commenting on the particular specifications these rules mandate for F1. Specifications that are not fit for purpose.
There's no reason the battery should run out on the straight, though. You can charge at any point while driving, not just during braking. Obviously you're taking torque from the engine to do so, but that's a strategic consideration.