Just to be sure: the wave function is still a distribution of complex amplitude over a configuration space, right ? If I understand, you are saying is that if we have a Dirac peak at some point, then the momentum spread everywhere (the infinite momentum). I don't dispute that.
I only meant that we could describe (in principle) a hypothetical wave function with infinite precision. Meaning, at each exact point in our hypothetical configuration space, we would specify an exact amplitude. (With the usual caveats due to the fact that we're talking about a distribution, not a function.) From then, we could predict the experimental results, which are bound to yield finite precision.
Now, current physics say we will never infinitely accurately measure our wave function. But it also says that this wave function behaves lawfully, with infinite precision, from some (I think?) unknown initial state. Is there a problem left ?
Edit: it just hit me that there is a problem if you don't believe in Many Worlds. Just know that I think the Copenhagen interpretation is crazy (I don't believe in the collapse of the wave function), and that currently, I find some form of Many World Interpretation most likely.