Former Sleeman's Employee, worked with a microbiologist that had spent something like 30 years at Molson.

As you said, Canadian prohibition was never enacted in many of the markets of Canada to the extent of prohibition in the states -- Quebec's prohibition was repealed the same year it came into effect because of public pressure. Additionally, because federal and provincial regulations were different, oddities occurred like wineries and breweries in Ontario remaining open for export markets for a time (until WW1 stopped all sales). So, this meant that many of the breweries in Canada were still operating during the prohibition.

However, this did mean breweries started falling on harder times (just as what happened in the US), because the US was a massive, now illegal (but still profitable, see Sleeman's) market that most companies had lost. Many breweries DID just close, being unable to sell to their local markets. During this period, something like 3 out of every 4 breweries in Ontario closed. So, to answer your main question, the reason Molson and Labatt rose to prominence was essentially the same as the reason that Bud, Miller and Coors did.

More in-depth, Molson and Labatt in particular success stories for two different reasons:

Molson is one of the oldest companies in Canada, and they've had their hands in many pots over the two hundred plus years they've been around, like so many businessmen. Brewing was their most successful business, but the Molson family also operated steamships, lumber yards, foundries, was involved in politics, etc. This tradition continues even now: The Molson family is the largest owner of the Montreal Canadiens. I don't know the specifics, but I expect they simply bought brands as they went bankrupt and had to sell, taking money from other businesses to keep everything in the black. Molson is also located in Montreal, which was easily the largest market in Quebec during the period, in the province that had the shortest period of prohibition (and in Quebec, like in many European countries, beer/wine were treated differently from spirits).

Labatt (while still making full strength beers for the US) made 'temperance' ales during the early years of prohibition in Ontario -- light (alcohol wise, not colour) beers that were legal to sell before the blanket prohibition that came during WW1. This extra income from Ontario before prohibition during the war fully banned alcoholic drinks probably helped them through the period when most of the breweries were going bankrupt, and let them come back stronger when prohibition was repealed with so much less competition.

As far as lagers as the dominant beer style, the reason lagers are the most common beer worldwide has a couple different reasons, and it's not only because the brewers who were lucky enough to survive prohibition were lager brewers -- ale yeasts, which ferment on top of the beer, are more easily contaminated because for those kinds of yeasts, fermentation is most active at warmer temperatures. These warm temperatures are also more conducive to wild yeasts and other microbes. Because of these "infections" (they're not always bad, and are in fact very important for some beer styles. If you're interested, look at Lambics/Gueze and Belgium Beer in general), they also typically have a shorter shelf life.

A single strain of lager yeast was isolated in Carlsberg laboratories (a name you might recognize) in 1881, and could be stored for long periods (in a method that is basically still used today), allowing a specific yeast strain to be purposefully shipped worldwide, one that had known effects on the beer and the flavours produced. This consistency was hugely important, because it meant that someone who liked Molson Canadian (or Coors, etc...) could always expect the same taste from the same brand (which is not true in something like a home brewer's operation, for instance), and more importantly, no bad tastes, since colder fermentation suppressed more of the microbiological beasties infecting ales (note that this colder fermentation is also one reason lagers take longer, but I'll talk about that somewhere else if someone asks).