It is often a case of having to set absolute rules because some people are always going to stretch what is "normal." If they say you can trim some branches, then some idiot is going to start doing major harvesting with the excuse that "well they say it is allowed" and then the park will have a hard time enforcing a citation in court because it was not specific. And then the next guy comes in and wants to trim branches on a tree on the other side. Pretty soon the campsite does resemble a parking lot.
If they flat out prohibit it, then the prohibition is clear. Unfortunately, it also takes the practicality out of all situations, even when a small branch is in the way.
So the specific answer is "No". You are not the park manager or park staff, whether it is a public or private campground. It is not your job to clear a site. You may feel they are not doing their job, but it is not your role to do it in their absence. You should notify them of the problem, and ask if they can clear it or if it is OK for you to do so.
I was at a USFS campground one time and a guy whipped out his chainsaw, stood on his trailer and started cutting away at branches way above his trailer. Then he whipped out a weed wacker and cut the ground vegetation enlarging the cleared area of the site to be about twice as big as it was. These are the kinds of people that ruin things for everyone because they have no "common sense" and don't care about anyone but themselves.