I was informed that modern day St Pancras is designed to be deliberately drafty as a fire prevention/smoke extraction measure.
Would't surprise me if same thinking has been applied at King's Cross.
I suspect that you are correct. Most deaths in fires are from smoke, not burns. Designing buildings so as to reduce the chance of smoke build up can save many lives.
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Why not time your arrival to train departure -10 by which time it should be boarding.
As others have posted, this is not always viable due to the risk of missing the train if delay occurs en-route.
My usual policy is to arrive in the vicinity of the station up to an hour before my train leaves., that is not to say that I stand around in the cold for 50 minutes before boarding however.
Possibilities include, waiting in a nearby pub or café, or if arriving by underground, taking a seat on the warm and sheltered underground platform.
At Paddington I normally wait in the First class lounge
At smaller or more remote stations there may be little alternative to a wait in the cold, all that can be done in such cases is to dress warmly and take reading matter, food and drink, or electronic entertainments.