Capa,
You are doing it right!!! Lighting the fire and driving off warms everything up together, i.e., transmission, axle, bearings, etc.. Also, your car will warm up faster with a load on it vs. sitting and idling.
Unless the choke was finely adjusted, many of the older cars with carburetors wouldn't take power until they ran for a few minutes ... maybe a lot of minutes ... before you could finally shift it into gear and drive off. And, if you had a manual transmission, you had to wait for that old molasses gear oil to thin out so you could shift gears!!
Modern cars ...no problem ... with fuel injection you should be able to start and drive off right away even on the very coldest day provided you don't have to de-ice the windows, etc..
Here's the controversy.... You should be able to light the fire and drive off right away even in really cold weather IF you are using a 0Wx or 5Wx oil. These oils pump and circulate very easily when cold so the engines' parts get lubricated right away. This is a good idea regardless of temperature... Some will argue that you need to wait till the oil warms and circulates before getting off. Not so.
I use either 0W or 5Wx oils year round. I just fire it up and drive off. A 65 mph highway is a long block away and I've no problems with jumping on it on a cold day ... even if the temp gauge is pegged at the bottom of the scale.
Stay warm!