Why the #### would there be a gate through the wall? If there's something bad enough that you need to built a wall that big to keep out then don't build a ####ing gate through it.
This. I realize we need to suspend disbelief for a story that has giants and dragons. But if you're going to somehow build an ice wall that's over 700 ft. tall and over 300 miles long, it sort of defeats the entire purpose if you also include a ground level door for it. The tunnel should be higher up the wall and have an elevator, just like they do inside the wall.
It's also makes no sense for the enemy to send any troops up the wall instead of attacking the tunnel with everything they've got.
There are a few problems with attempting to open the gate and attack through there.
First, the Crows could just continue to drop the bombs on anyone who attempted to open the gate. Eventually you may be able to break the gate, but at what cost. 5,000 men. 10,000 men. More? The wildlings aren't some close knit group. They could break at any time. Losing that many men to open a gate would be devastating.
Second, didn't you see 300? If you attack in that tunnel, your numbers mean nothing because the tunnels are narrow. Because the wildlings have no idea how many crows there are guarding the gate, you probably don't want to take that chance.
Third, because Manse was a former Crow, he had to suspect the possibility of them freezing the tunnel (as Jon Snow suggested). So you lose, a ton of men trying to open the outer gate only to find solid ice waiting behind.