My thoughts, and I have actually thought about this topic:

Barbed Wire fences can be gone through after dropping your pack to the ground, cross between the wires, and then pull your pack through, under the wires. No need to carry a Fencing Tool, IMHO.

Gates usually have chains and would require bolt cutters, which have significant weight. Gates usually close the gap between barbed wire fences; see the paragraph above. No need to carry Bolt Cutters, IMHO.

Cutting a fence or chained gate leaves a calling card broadcasting your presence. A decent tracker can tell how long you were at that location and which direction you were travelling in, and usually how loaded that you are, which then tells him/her how fast you are likely travelling. IMHO, NOT a good idea.

If you are "Bugging Out or Bugging In", advertising your passage/presence is never going to be a good idea, generally speaking, IMHO.

It is definitely a better idea to adopt the US Navy SEAL atitude of moving like a Ghost and leaving as much of a trail as a Ghost!