"What trait in dogs, in overwhelming numbers, is the main reason that dogs are killed in the UK?"

It's a trick, leading question that banks on you making a variety of assumptions

For clarity the answer refers to an INDIRECT cause and to get it you'd have both needed some out-of-the box thinking and a similar belief system to ours/mine

I'll explain my belief system that may alter your perception (or may not)         before I give my answer 

I don't believe in "domestication" as being some kind of achievable target.         I believe that most issues in dog behaviour (25% of all dogs) come from the dogs inability to cope with the demands of modern day "domestication", the other 75% are so eager to please their humans that they let us get away with murder at the expense of their own needs and instinct

In addition, it's not the openly aggressive dogs that are the most dangerous ones because you see it coming and are prepared for it. The most dangerous dogs are the dogs that are 99% fine but in the other 1% their aggression is at a high level

These two factors lead to us taking advantage of the dogs nature when it comes to us using them to our own end and then being massively complacent when it comes to any bad behaviours and understanding them

Therefore the answer to the question and the dogs own achilles heel that leads to most dogs being killed is not a NEGATIVE trait in the minority it's a POSITIVE trait in the majority

It's their TOLERANCE. Dogs are SO TOLERANT & EAGER TO PLEASE that it's massively detrimental to their species

The DR philosophy and belief system is about rebalancing our dogs lives by   removing the pressures and expectations that come with modern day             domestication and swinging the pendulum back firmly to meet their needs as a species and rolling back the clock in terms of human interference and involvement