never understood why a second runway at Birmingham was never floated.
Because it's not in - or close enough to - London.
There are other technical reasons too, but on a fairly fundamental level, infrastructure spending is only politically justifiable if London gets the majority, or all, of the benefit.
To be sure, there are some justifications - London and the surrounding commuter belt represents by far the highest (productivity x population density) product in the country, so there are correspondingly greater returns on such - but no politician is ever willing to actually discuss the chicken-and-egg problem inherent in that observation, or to acknowledge that it is a compromise - but nevertheless, that is the way it is (and why most of the non-London regions tend to hold the south east, and London in particular, in at least slight contempt).
It was obvious when, e.g., HS2 was started from the London end instead of from Manchester or Leeds, and split into multiple legs - so that the additional legs could be sacrificed due to the entirely predictable budget overruns. It was obvious when, e.g., railway electricifcation largely stopped in the 70s after the main lines to London were complete. It was evident when every aspect of the Northern Powerhouse was quietly dropped as soon as the Tories won in 2019. It's why the planned expansion of the Newcastle Metro system was mysteriously cancelled in the early 1980s, and no other city in the country (other than Glasgow, which predated the London-centricity by about 30 years) has ever gotten the funding to build something similar, while the London Underground has had three new major lines added since then and another planned.
I could go on, but you get the point (and probably also perceive a certain railway-related theme to my interests).