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Canada doesn't share a land border with any Schengen countries, so I'm not sure how valuable passport-free travel would be - you're already taking a transatlantic flight, it's not like passport control is a huge additional hassle.

However, I could see a big argument for a labour mobility agreement similar to the NAFTA model, where EU citizens in a long list of professions have the right to work in Canada and vice-versa.

It says a lot psychologically to take a 'domestic' flight instead though, gives people more of a sense of togetherness

The Council of Europe is strictly speaking not an EU thing at all, though all EU countries are required to be members of the Council of Europe, and, confusingly, they share a flag. The UK is still a member of the Council of Europe, for instance, as was _Russia_ until 2022.

Schengen doesn't have anything to do with currency. You might have been mistaken it with the European Economic Area?