Dirac delta being zero almost everywhere is sometimes assumed (when describing infinitely short impulse) but often it is a superfluous assumption, that is not n

Dirac delta being zero almost everywhere is sometimes assumed (when describing infinitely short impulse) but often it is a superfluous assumption, that is not needed for the use of delta. This is because delta is a distribution, not a function. It does not need to be ascribed values. For example, delta can be put as initial condition for psi function in the time-dependent Schroedinger equation to search for Green's function. Evolution of this initial condition in time is a regular oscillatory function that looks nothing like 0 or almost 0 almost everywhere for any $t>0$. So if it isn't 0 almost everywhere for t>0, why should we impose that at t=0.