It says you can leave the engine on, providing the handbrake is applied, to “drive machinery forming part of, or mounted on, the vehicle and used for purposes other than driving the vehicle”
It doesn’t say that, unless you’re choosing to read it selectively and ignoring the bits inconvenient to your argument.
(1) Save as provided in paragraph (2), no person shall leave, or cause or permit to be left, on a road a motor vehicle which is not attended by a person licensed to drive it unless the engine is stoppedAND any parking brake with which the vehicle is required to be equipped is effectively set.
is what it actually says.
And 2b qualifies the exceptions in 2bi and 2bii with in such a position and condition as not to be likely to endanger any person or property
So the additional exceptions in 2bi and 2bii only apply if the first part of 2b is satisfied. Good luck convincing a police that 2b is satisfied if parked on a public road or in a public place (e.g. car park)
How does that work? Drive chain is part of the engine so it would warm up as the same rate as any other part, except maybe the exhaust/manifold and parts near it, which would warm slightly quicker.
The plus side of a 400m walk to the car is that it can be put into defrost mode at home, and be ready when you get to it. Remote access is quite handy.