I have also been at a larger company, so you bring up a good point about fighting for work. I also think this is right.
1. If you win the fight for high visibility work, that's a good sign that you're the right person for the job. It wouldn't make sense for the company to have you work on something for your own benefit over the company.
2. You may opt not to fight and discover other problems to work on that have strong potential for being high visibility. Maybe that's something relegated to PMs or eng management, where the engineer might feel helpless to the given assignments, but that's more of a culture problem. I think politics increases when there's less work/budget/praise than there are people, which is more situational than cultural, in my opinion.