If Canadian forces have "traditionally" fought for the "honour of the Regiment", and the US system of "numbered units" was "inferior", you'd have to explain to me why the Canadian Expeditionary Force was so overwhelmingly successful in the First World War. Aside from a few kilted units and some larger Militia units that managed to preserve their pre-war identities in the form of regimental insignia, the CEF was composed largely of "numbered units" with little or no history...and fought quite well.
"The US system is based on blind obedience by large numbers, while in contrast our men are traditionally taught unit pride, personal self discipline, and how to think for themselves on the battlefield as a part of a smaller organization."
I would argue that the US system also taught unit pride, personal self discipline and how to think for themselves on the battlefield as a part of a smaller organization. You don't need funny hats or metal cap badges to do that. Band of Brothers by Stephen Ambrose, or any decent small unit history, will emphasize the role unit level training plays - in any army. It was no different in the Canadian Army. In fact, if you read, for example, Farley Mowat, he talks far less about the "honour of the Regiment" than he does the personal relationships among the men of the sub-units.
The regimental system is a nice thing to talk about, but could be easily jettisoned for an "American-style" numbered system (and, really, we shouldn't call it American, because the Germans, Soviets, etc. also used similar non-British systems of organization) in practice and not suffer too greatly - how do I know? Because we helped win a world war by doing it once.