I don't know why this is the case, nor can I probably conclusively prove it as it's a hard thing to describe and there's many different conditions in each of the cities. Best I can do is my experience walking as pedestrian in the urban centres of all these places. Compared to Vancouver, Toronto and Montreal - all with far more pedestrian culture, more rain, and in Montreal's case, far more snow, ice and slush - Calgary and Edmonton feel they have way higher curbs and more "steeper" road cross-sections.
I find this phenomenon pretty much unnoticeable while driving. As a driver, the only time you'll notice is how sloped your car is in the curb side lanes on places like 11 Ave or 12 Ave SW. There's other weirdness that's a clue - sometimes the bottom of your door scratches on the sidewalk when you swing it open, pop-up patios have weird steps and grade changes need to make a level floor on an uneven road surface etc.
Where it becomes noticeable is when you spend enough time walking around the pedestrian cores of all these cities - particularly when crossing the road at a right-angle, such as a crosswalk. If you walk everywhere even in Calgary's most walkable areas, you'll start noticing it's fairly regular experience to see or be required to help a wheelchair user in the Beltline to roll out of the gutter as the drop from the curb is steep, followed by a steep climb to the centre of the road. Same goes with strollers and other carts, and folks with other mobility supports.
This probably doesn't make much of a difference to most able-bodied folks. But that's missing the whole point - everyone will be in situations in life where the have their mobility restricted at some point temporarily or permanently (strollers, carrying groceries, injury, illness, age etc.) why are streets not designed to to acknowledge this? A pedestrian-friendly city acknowledges the attention to details really matter for pedestrian accessibility. An extra inch or a extra step really does matter in many situations to pedestrians.
I think it's clear that drainage designs for road profiles are the source of the issue. But it isn't sufficient alone to explain the variability of that standard in application, nor why rainy/cold/slush/ice covered cities seem to be fine with lower profile curbs and less curved road cross-sections. Montreal seems perfectly happy have some streets with zero curbs at all, including ones with regular vehicle traffic.
By comparison, Calgary's inner city streets and sidewalks are all over the place in street width/slopes, sidewalk width/slopes, curb height, pedestrian ramp angles/quality/condition, amount of curb cuts, sidewalk closures, random debris, random poles locations/widths. In most cases the randomness is not some thought-out trade off between drivers and pedestrians - but a product of a culture of sloppy construction or general ignorance to the pedestrian experience and the attention to detail it requires. Some of these are annoyances but not overly harmful, many actually are anti-pedestrian and go against the basic right to mobility for everyone.