Understanding by symbol pushing is not real understanding. This is all that is wrong with how mathematics in college is taught. They start and end with symbol pushing, and the real insight remains locked in the professor's head. The professor projects his insight into symbols, and the students only learn the projection.
What should be done is start with the intuition and visualization, and then show how the ideas can be made precise. If you would have been taught only the epsilon-delta form from the start, it would have been easy precisely because you would not yet have obtained the real understanding.
Even in high dimensional spaces visualization is very important. This often happens by analogy with lower dimensional spaces. For example if you start with a region in Hamiltonian phase space, then the region's volume is constant as time evolves. This is a highly intuitive and visual statement. Saying the same in symbols wouldn't be nearly as clear.
What should be done is start with the intuition and visualization, and then show how the ideas can be made precise. If you would have been taught only the epsilon-delta form from the start, it would have been easy precisely because you would not yet have obtained the real understanding.
Even in high dimensional spaces visualization is very important. This often happens by analogy with lower dimensional spaces. For example if you start with a region in Hamiltonian phase space, then the region's volume is constant as time evolves. This is a highly intuitive and visual statement. Saying the same in symbols wouldn't be nearly as clear.