This isn't a dichotomy. My point is that there are clear examples of situations where you aren't just pushing complexity around, but actually achieving great simplifications.
No it wouldn't. The complexity of a pattern can usually be conserved while reducing its length, but for each pattern there is a limit. This is the entire concept behind the Kolmogorov complexity of a system and any patterns that cannot be reduced any further without removing complexity are at their limit already.
This is also related to the idea that you cannot have a universal compression algorithm.