It may be public land but the gate is someone else's property and it's not "the public's" responsibility to maintain it.
It's not the public's responsibility to break it either, but they do. Or leave it open so the cattle get out, but they do.
Or as evidenced by the shotgun mark, and broken hinges in the OP photos, to shoot the latch chain off, or just drive through it to open it...but they did.
Or shoot up windmills, water tanks, and cattle troughs. But they do.
Or shoot an expensive registered Angus bull in my pasture. But they did that too.
Most any newish fence/improvement will have a maintenance agreement with the permittee. If funds are available they could be some cost sharing of that, as mentioned above. Sometimes not. Other eras of range improvements the maintenance is on the agency as that was the policy of the time. Even others are wholly owned by the permittee, including the maintenance and this ownership goes with the permit. So, it's murky at best...