Higher level math requires abstract thinking. Abstract thinking is hard. If you can learn to think abstractly then it will allow (or help) you solve a large class of abstract problems. If math is broken down such that it does not require abstract thinking and a person does not learn to think abstractly then they will not be able to solve that large class of abstract problems.
Or so my line of likely incorrect train of thought goes.
I agree that modern technology may help provide better representations, but in the case of most of abstract mathematics, I am not convinced it currently does, or even that it can. There is some truth in the statement "a picture lies more than a thousand words"
Not necessarily. There are such things as inherent complexity and accidental complexity. forrestthewoods's point was that some things are inherently complex, which I agree with. Accidental complexity happens when we use a poor representation - such as roman numerals.