I appreciate that the stated aim of the policy is to improve the safety of the workforce. However, if a bin is overloaded or too dangerous to put into the van, the workforce can take other measures to resolve the problem - like removing a bag from it before loading it.
I appreciate that there's also a case to be made for people trying to get rid of too much rubbish, which is why you have a service to sell overflow sacks, to ensure that people who have more rubbish to dispose than expected have some way of doing so. However, a bin's lid is more dictated by the volume of refuse than the actual mass being disposed of. It only takes a few funny shaped items to make a light bin hard to compact into one with the lid down.
If the policy were applied with reasonable common sense, and a polite request were left on an empty bin, asking for the policy to be followed next time, then there wouldn't be as much of an issue. The fact that the bin-men flat out refuse to collect a full bin is the exact opposite of what they should be doing.
The fact that there have been fatalities within the industry is very emotive, but I think it's not worth creating a dogmatic illogical system over.
So this is my challenge. Justify the policy. Give me the best argument you have for what it's reasonable and the right thing to do. If I can convince you that your argument is not logical, then you can repeal the policy, replacing it, as I suggested above, with a polite note, asking for the bin to be closed next time, rather than the patronising and jobsworth-like tag that gets put on a bin that's not to the bin man's liking.
I've fallen foul of your policy on two or three occasions. The last time was a result of my brother in law putting our bin out for us as a favour and assuming that a 4 inch lid-ajar situation wouldn't be too bad. Now I have 4 weeks worth of rubbish to put out next time. It's plain silly. Let's see if we can discuss this logically and get some common sense back into the picture.