Americans think that Canadians pronounce about as aboot (I've never heard anyone pronounce it that way) yet they pronounce route as root. They know how to pronounce out, about, router (as rauwter) but then why do almost every American pronounce route as root and they don't see the double standard?
If you look at forvo.com, all the Australians, Brits, and Canadians pronounce "route" as root. It is only around half the Americans who pronounce it as rout. So the historical British pronunciation is presumably root. The historically incorrect rout pronunciation probably originated in America.
I think it's probably two things. First, an exaggeration of an actual difference the pronunciations, more noticeable in some people and to some people. And second, others perpetuating this exaggeration just because they've heard someone else do it, not necessarily because they've witnessed this phenomenon themselves.
I find it quite easy to distinguish Canadian speakers from American. There are a lot of subtle differences between the dialects, which jump out at me. 'About' is one of the 'red flag' words. Others include 'object' and 'sorry', which, to exaggerate a bit just for demonstration, sound like awe-bject and sore-ee to my ear.
There's a children's show on PBS and CBC (with Canadian actors) called 'Super Why' and at some point in every show, the main character says, 'Super job, Super Readers!', and, to me, it sounds like 'Super Jaaw-b'.
It's because of the way French pronounce their
aʊ sounds when speaking English, sometimes. Maybe some French Canadians go in and out of a strong French accent when speaking? I don't know many French Canadians, so I can't be sure, but it's true about French people. Their accents are a little powty, and the word 'about' does sound a lot like 'aboot' when the accent is strong. It's perpetuated in South Park, and this is where I believe it originated from. I agree that Canadian accents aren't like that.
I'm pretty sure the pronunciation stereotype originated from the Bob and Doug McKenzie sketches on Second City TV. Note that both of these characters were played by Canadian actors -- Rick Moranis and Dave Thomas.
There certainly are Canadians who pronounce 'about' like 'aboot'. I've worked with them at a company in southern California. It was particularly clear when they had just returned from a trip back home. The people in question came from New Brunswick, Newfoundland, etc. They weren't French Canadians either, although I know at least one spoke French.