> a warm drink does little to improve your core temperature.
This is true, but there is more to it than that.
Imagine that you are a bit cold, or maybe low on energy. Your body's reaction to this will be to restrict blood flow to your skin and concentrate it in your core, to make sure you stay alive. This is good of course, but it leaves you feeling cold.
Put a good mug of quite hot drink inside you. Now your core thinks it is a just a bit hotter than it should be, so it switches a lot of your blood flow outwards to your skin, even if just for a short while. This increased blood flow carries heat to your skin to get rid of the heat from your core.
But you don't really have a lot of temperature sensors inside your body: most of the ones you are conscious of are on your skin. So now you start to actually feel warm, because there is increased blood flow at your skin. So, deluded by this false feeling of warmth and joy, you leap out of bed into action.
Yes, both my wife and I start dinner with hot soup, and may have a hot drink with breakfast in winter too. The only problem - I have to get up to fire up the stove... :-)