I don't think Rome ever had any grand scheme for conquering the world. More that they kept going until it cost more than it gained to keep going.
I think Rome initially moved north because Rome had repeatedly been attacked from the north. It was a matter of time before they were overwhelmed if they kept playing defense on that border. It may be that they couldn't afford to stop once they started.
I'm not partisan in terms of who was right or wrong. By our standards, there were no good guys back then.
I concur, the conquest of Gaul was popular because of the Gallic invasions of Italy, during one of which the Gauls sacked Rome. By Caesar’s time that was long of out living memory, but the Romans still remembered it. :’)
Caesar’s successors had to deal with occasional incursions, but mostly managed their frontiers through the discipline of the legions, a few long walls, and (mostly) stabilization of tribes and other groups just over their borders, iow, creation of buffer states.
One problem was the devastation of Gaul wrought by Caesar — as much as half of the Gallic population was displaced, and a significant fraction was carted off into slavery. The vacant lands were discussed and drooled over by tribes east of the Rhine. And that problem went on for centuries. It was finally resolved when the thin bronze line failed for good, and European populations familiar today started to enter and take root.