Yes.
Are there then any production methods that this vast majority would allow to happen that would change their eating habbits?
My opinion on this as a vegetarian:
No matter how it was produced, I would not eat meat because I personally like animals more than I like the taste of meat. If my life depended on it, I'd eat meat, but not so long as it is optional.
I have no problem with other people eating meat though. I cook meat for my husband.
However, I have a big problem with the cruelty and environmental hazards posed by large scale industrial farming. I purchase free range meat, eggs, and dairy from local farms. I am hugely in favor of laws which would regulate the way factory farms operate to provide certain environmental standards and more humane conditions for livestock. I am in favor of this even if it causes the price of animal products to increase. Especially because Americans as a whole eat far too
much dairy and meat. Both dairy and meat are not inherently unhealthy, but they are in excessive quantities. I had participated in successful campaigns in Florida in which we have gotten questions on the ballot which improved conditions for farming for animals that were voted into effect. One of the reasons these campaigns were successful is because vegetarian groups formed coalitions with small, family owned farms, which did not have such excessively inhumane conditions in the first place (unlike factory farms).
Just thinking; meat being produced in a humane way wouldn't make me any more likely to eat meat. However, I personally do know people who are not vegetarians, but who will only eat humanely raised meat or meat obtained from hunting.
I can't say what percent of vegetarians that includes, I've never looked into it. What I have found in my limited experience is that how strict a vegetarian is depends on their age/length of time they have been a vegetarian. Many vegetarians I know became vegetarians at a young age, and started out being strict vegetarians or even vegans. However, over time (particularly if they married non vegetarians) they started to eat humanely raised meat, though pretty rarely. In college I knew a lot more vegetarians when most of my friends were in their teens and early twenties than I do now when my friends are in their late twenties to early forties. Now I know a lot more former vegetarians who eat small amounts of humanely raised meat. I'm actually one of the only people I know who was a vegetarian ten years ago and still is today.
But on the other hand, I know quite a few former non discriminating meat eaters who have greatly decreased their meat intake for health reasons. This I think is not so much due to vegetarian campaigns as to campaigns in recent years from medical associations like the American Cancer Society, American Heart Association, American Diabetes Foundation, etc, on the importance of a low meat high plant diet. In addition to public education campaigns, I work in health care, and now in large hospitals it is fairly standard practice to recommend nutritional counseling for people with chronic diseases (or who are at genetic risk for diseases like cancer) to explain to them how a diet low in meat and processed foods and high in plant material significantly decreases their risk factors.
It should be noted that I do live in Boston though, the medical provider/research capital of the nation, if not the world. An enormous number of people in the area work in health care or in health care research or know a lot of people who do. So I live in a population which is probably much more informed about health and nutrition than the general population.