Having thought about this, I think we should break the question down.
First, there isn’t a “Great Wall;”, but a variety of wall-based defenses that different rulers placed at different times. (For that matter, there isn’t “a” China, but a lot of different kingdoms and regions that were integrate to greater or lesser degrees and at different times. But OK, that’s really nerdy.)
Second what was the purpose of building these defenses? Actual military defense? Border definition? Slowing raids? Showing the power of the state? Wiping out inconvenient populations through forced labor? The Wall kinda-sorta did all of these at some times and not at others.
Sure there were gates. But the wall was never defended like you would defend a city wall. It was garrisoned with a force strong enough to watch for bandits, and to keep armies from just climbing over the walls with a few ladders or tearing down the wall. But a proper army could go over/through the wall without much trouble, it would slow them down for mere days. It wasn’t like having to either besiege or bypass a fortified city. It was never intended to stop invaders cold because it could never be manned to the level where any one section of wall could repel an invading army. It was more like a system of mutually supported watchtowers with palisades between them such that no army could cross the wall without word being sent to the central government.
In addition to creating a speed bump for invaders, I understood the wall was also built to keep a portion of the populous busy. With all those people, they needed jobs, jobs, jobs, so it was a hundreds of years public works project motivated by fears of the people on the other side. Although, I doubt China got the Mongols to pay for it.
As said earlier, the Great Wall was designed to be an obstacle for attackers; a place which would hold them up and hopefully cause heavy losses degrading their fighting ability. It was not supposed to be an impenetrable barrier.
People mix up border fortifications, like the Great Wall with City Walls and Castles. The later *were *supposed to be as impenetrable as possible. The reason is obvious, city walls get breached the game is up. You *expect *border fortifications to get breached or bypassed
And remember, you may be able to get past the wall and few complacent defenders to do a little raiding but, when you turn around to go back home, the wall is still there and the defenders aren’t so complacent now.
It probably worked as well as any city walls worked - stopped most, deterred small attackers, and stopped or seriously delayed bigger armies.
Also if you look at pictures, in the areas most needed, near Beijing etc - it was mostly run along the top of ridges, with the occasional descent across a valley. So attacking anywhere except the obvious valley gates meant attacking up a serious uphill.
In addition to what has been mentioned above, there was also the psychological impact. An invading barbarian might well pause and say, “Do we really want to attack a country that can build something like that?”
Yes, but the people it was *supposed *to keep out conquered China in the meantime. Had an imperial dynasty and everything. At best, the wall stopped or slowed down opportunistic raids by unorganized bands. As has been already said, it was also a land claim of its own. Whether that proved cost effective is up for analysis.
What is clear is that the cost of constructing, maintaining and manning the wall forced China to stop its erstwhile exploration voyages and building up of commercial ties around the Indian ocean, which a) slowed its economy and b) allowed the Portuguese and the rest of the Europeans to corner the markets & seize key ports themselves and then find America. Obviously the Chinese couldn’t factor “loss of opportunity re:undiscovered continent” into their decision to build a wall, but in hindsight it’s kind of a big deal because they had all the tech required to cross the Atlantic or Pacific at least a good century before Europeans ever did.
Nothing. Just goofing around and making a point that the Chinese could man the wall. Not saying it made sense or was the wise thing to do but it was technically possible.