This is potentially a rabbit hole, but it's worth exploring. To begin, I agree - it's sometimes a bad proxy for measuring the quality of driving, but generally it's a pretty good one. I grew up in Belfast and as a side effect of enthusiastic and well-resourced policing, driving and vehicle standards were much more rigorously enforced than I've ever seen in England.
Speed is a common factor in a lot of accidents and prima facie, if something goes wrong and you're going fast, you have less time to react, correct and avoid a bigger problem. Speed can be easily measured against an objective limit and thus you can have a simple measure of compliance with the law. So you have something which is easier to measure and enforce. If someone is confident enough to think that they can safely drive faster than the speed limit, I'd argue they're more likely to think that they can drive and text without a problem, or overtake others in dangerous circumstances.
More generally, and keeping it prima facie again, speed is also a common denominator in lots of different types of bad driving (dangerous overtaking, for example) and it's easier to get a conviction with something which has an objective strict liability test (speeding) than something like dangerous or reckless driving. The problem with pulling out without looking (as an example) is that it's a very difficult behaviour to capture.
Personally I think education should go hand-in-hand with enforcement to improve driving standards and make roads safer, for drivers and all other road users.
As a side note, I'm not bothered whether the 44mph man was in a Bentley or a banger - arguably he'd be more inconvenienced if he had an inexpensive car to lose because it might be all he can afford. The rich man could probably afford to replace his Bentley (though doubtless there would be a fit of righteous indignation).