One of the reasons that shoes smell is that the moist air inside gets trapped with a lot of sweat and bacteria which feed on its organic constituents.
In the summer, you could get away with wearing open-toed sandals (if you weren’t any kind of fashion victim) or at least light shoes with ventilation holes. No-one wants to wear these, though, when there are icy puddles everywhere.
Today’s invention is a breather tube fitted to each shoe in the otherwise unventilated region of the toe box. This would emerge from the shoe and snake around, attached to the outside, until rising to the level of at least the tongue. You could still step in puddles without getting inflow onto your socks and keep your feet aired at the same time.
Rather than rely on low pressure air flows, driven by just toe movements, the pipe could be attached to a small air pump on one’s belt or even a bladder within the sole which would compress under bodyweight, forcing air into the shoe and then spring open when unloaded, sucking it out again.