>Photons are said to go through both slits because they travel like waves and actually go through both slits.
not really. See mentioned below Hitachi electron double slit experiment.
> How would your interpretation account for the fact that a single photon at a time fired through a double slit still produces an interference pattern?
Because position is quantized, ie the position probability has that wave form, and thus position probability of hitting the screen has that interference looking pattern. There is no real physical interference though between any real waves here. It is just a position probability pattern formed as direct sum of 2 other patterns - the 2 cutouts made by the slits from the original wave pattern of quantized position probability.
The Hitachi electron double slit experiment (https://www.youtube.com/watch?v=PanqoHa_B6c) is much more illustrative because electron position quantization is different from the electron's DeBrogile - unlike photons. In Hitachi you can see that each electron hits the screen as a singular point and only the statistical aggregation of these hits - such aggregation naturally visualizes the position probability density - forms the "interference" pattern.