How important is the function of the pre-heated air tube from the
exhaust manifold into the air box?

I can't give you all the reasons why old cars have heat stoves of one sort or another, but I can give you a few: Carburator icing is probably the biggest reason, something I've never experienced in Alberta winters but we don't get a lot of humidity. Carburators/distributors and primitive ECU/ICUs can better do their job when as many variables are made consistent, and this includes air temperature. The engine likely makes slightly fewer emissions and fractionally better economy with the heat stove, and the engine makes a lot more consistent power while the engine or while the day is warming up. Overall it feels more "quality" with it in place, but guaranteed removing it will add power even on the hottest days - on cooler days the engine feels like it's got another cylinder or two.