Differential equations and LTI systems
Your "mistake" if it can be called that is to assume that your input signal was zero for $t\le0$.

When talking about the ZI-response and the ZS-response of a (typically causal) LTI system, "zero input" means "zero input from now on".

In that case, the past input's influence on the future signal is fully expressed with the ZI-response, while the future input's influence on the future signal is fully determinable by the ZS-response.