Apologies for typos.

Well this is a more completed question but the basic answer to your question is yes, sort of. Take a look at this map.

As you can see, there wasn't really a "Great Wall" but rather many, many walls that are collectively Great Walls. The building of the Great Wall was begun by Qin Shihuangdi during the Qin Dynasty (221 B.C.E. to 206 B.C.E). However, several states engaged in wall building which can also be considered part of the Great Wall. The bit of the Great Wall that you see in all of the pictures is usually the very impressive sections that were built and maintained by the Ming Dynasty (1368-1644).

This is important because it should communicate that the Great Wall or walls were designed to meet different needs for different states and often had different purposes in different areas. Fundamentally however, the Great Wall was not wholly to "defend" China. This was an enormous set of structures and required massive numbers of troops to fully garrison and make effectively defensible.

Think about a castle, even the largest castles took a substantial number of troops to directly defend a small area. For the wall, fully garrisoning it like a castle would be challenging and expensive. If an invading army wanted to, it could easily pick a section of wall and quickly scale and overpower the local garrison.

So what's the purpose of the wall then? Well the wall, or walls or fences in general are more about making it more difficult to pass than actually stopping cross border movement altogether. If you have an iron link fence with a gate around your house, you're probably not under any illusion that the fence will stop an intruder from entering. However, it will make it harder if a burglar wants to enter. He or she would have to climb or cut the fence, then carry the stuff they want to steal back across or through. If you take this, make the fence much higher and imagine hundreds or thousands of raiders doing this, then you can understand how the walls on the northern frontier functioned.

The nomads to north of China, whether Mongol, Jurchen (Manchu), Xiongnu, Khitan etc. (the names and ethnicities of these various groups are hard to pin down) subsisted on hunting, husbandry, and raiding. While a large army could relatively, easily cross the walls and invade China, this required a substantial amount of resources. If a party of raiders simply wished to invade, loot, and return, the wall made this much more difficult and slower. This was further facilitated by the fact that the wall enabled Chinese military forces to be warned and gave them more time to respond and hamper the efforts of the raiders.

Of course this was not the only function of the walls. You are correct in a sense that the walls were an important transit system of troops and supplies. However, not all sections functioned this way. Many sections of the wall were not as large as the more iconic Ming walls, especially the farther back you look.

I highly recommend the article "The Archaeology of the Great Wall of the Qin and Han Dynasties" by Xu Pingfang to help you understand how the walls functioned early on. The wall, which was constructed to connect many pervious walls built by other states, was between 4 and 5 meters high and at some sections 3 meters wide. Relatively large, but many sections were certainly not as large or well-built because of their age. It's unknown how many men were needed to garrison the wall. Contemporary sources stated the Qin used 300,000 men to invade the north and "drive back" the Xiongnu, this is likely an inflation of numbers. Moreover, the standing armies of these states was very small and often divided between various lords during the "feudal" period of Chinese history. Feudal is used here somewhat problematically but that's another discussion. In short, we have very little idea what kind of border garrison there was or wasn't at the walls.

Another important feature of the walls, which would, become increasingly significant, was as a means of border and trade control. The wall, even a relatively low section, made it necessary for merchants through set points if they wanted to access Silk Road centers past the wall. This meant that the states and polities that controlled sections of the wall could collect tariffs and taxes trade going in and out.

What I'm trying to underline here with the information above, is that the wall different in composition from place to place. Whether because of the state's that controlled and their resources (ability to maintain, refurbish, and garrison) and by the geography of the area. You're right in the sense that the wall was a means of transporting soldiers and supplies, but it had many functions. The chief function was a means of complicating nomadic raids, but the wall also useful in terms of commercial control. This actually holds true for the Ming Dynasty, although the Ming Dynasty had a very large standing army, unlike earlier Chinese empires. Of course, the wall did not save them from the Jurchen invasion.

Edit: another great source for getting started with these sorts of things is Sun Tzu's Art of War. Military tactics and discussions aside, many editions contain prefaces that discuss the nature of armies and equipment during the Zhou and Qin empires. I remember when I was a kid and that was literally the first book on Chinese history I'd ever read. I was blown away by discussions of equipment, particularly the raising and equipping of peasant armies. It should not be understated how poorly many of these "soldiers" were equipped. While the Qin and other states had relatively advanced weaponry like crossbows, chariots, and iron swords, many peasants were simply armed with makeshift weapons like spears tipped with bone.