Why Do We Need Initial Conditions to Solve PDEs?

Question

I am looking for further clarity on why solving PDEs without any specified initial values was not "good enough." For example: say we had the ODE \begin{equation} y' = y \end{equation} without specifying any initial conditions. Solving it, we get: $$ y(x) = Ce^x. $$ Doesn't this tell give us an explicit description of the solutions to $\textit{every}$ initial value problem posed with the above ODE? Doesn't the end product of $\textit{specifically}$ solving the initial value problem give us less information than solving the PDE generally? If so, why do we care so much about IVPs? Why not always solve PDEs first and then think about "Oh I wonder what would happen if it started here..."?