You can postulate what the reasonable electron w.f. is. And this is how it is done.If the source is not known in detail you don't know the electron's original wave function.
If the wall is not known in detail you don't know the electron's wave function before the detector.
If the detector is not known in detail you cannot know where the spot is produced.
I'm not even sure that we can speak about the electron's wave function as it is probably entangled with both the source and the wall.
Notice, that if one has a problem with guessing the electron's w.f. then somewhat bigger
issue arises if it comes to the w.f.s of the detectors, walls, etc.. There, one
has to deal with 10^25 or more atoms. How do you propose to determine details of w.f.
for those objects?
The only realistic way is to guess their wave functions.
Why bother then? Since heavy guessing is inevitable, why not to guess just the electron w.f. and focus on its properties?
Otherwise you fall into a circular reasoning:
1. To determine w.f. of an object you have to measure it.
2. You don't know what you measure if you don't know the w.f. of the measuring
3. So, you have to measure the w.f. of the measuring apparatus, and so on ...
I propose to cut this circle in the most convenient point i.e. the one that requires
the least amount of guessing.
Would gladly hear about another way out, though.