Trapping is actually less efficient, more time consuming, expensive and can cause significant distress to the animal. Not to mention collateral damage on non pest species that get caught in traps.
Game meat contamination is minor and is really only a problem with the very young, pregnant women and those with impaired immune systems. It's a far smaller problem than salmonella and contamination in industrial food.
The Department Agriculture and Rural Development has a [URL="http://www.dardni.gov.uk/ruralni/pests_series_cmb.pdf"]decent pdf[/URL] on the subject with the pros and cons of traps and shooting and so on for various pests.